6.13 Summary and Conclusions

Afactor model is a decomposition of the returns ofsecurities into two categories: (1) a set of components cor-related across securities and (2) a component that generatesfirm-specific risk and is uncorrelated across securities.Thecomponents that determine correlations across securi-ties—common factors—are variables that represent somefundamental macroeconomic conditions. The componentsthat are uncorrelated across securities represent firm-specific information.

The firm-specific risk, but not the factor risk, is diversi-fied away in most large well-balanced portfolios. Througha judicious choice of portfolio weights, however, it is pos-sible to tailor portfolios with any factor beta configurationdesired.

The arbitrage pricing theory is based on two ideas: first,that the returns of securities can be described by factormodels; and second, that arbitrage opportunities do not ex-ist. Using these two assumptions, it is possible to derive amultifactor version of the CAPM risk-expected returnequation which expresses the expected returns of each se-curity as a function of its factor betas. To test and imple-ment the model, one must first identify the actual factors.

There are substantial differences between a multifactorAPTand the CAPM that favor use of the APTin lieu of theCAPM. In contrast to the CAPM, the multifactor APTal-lows for the possibility that investors hold very differentrisky portfolios. In addition, the assumptions behind theCAPM seem relatively artificial when compared withthose of the APT.

In choosing the multifactor APTover the CAPM, onemust recognize that the research about what the factors

are is still in its infancy. The three methods of imple-menting the multifactor APTare more successful than theCAPM in explaining historical returns. However, it ap-pears that firm characteristics such as size and market-to-book explain average historical returns more success-fully. Until we can better determine what the factors areand which factors explain expected returns, the implica-tions of APTwill be fraught with ambiguity and willlikely be controversial.

If the CAPM and the APTdo not hold in reality, the the-ories may be quite useful to portfolio managers. Recall thataccording to the CAPM, the market compensates investorswho bear systematic risk by providing higher rates of re-turn. If this hypothesis is false, then portfolio managers canmatch the S&P500 in terms of expected returns with a farless risky portfolio by concentrating on stocks with low be-tas. The evidence in this chapter suggests that in addition tobuying low-beta stocks, investors should tilt their portfo-lios toward smaller cap firms with low market-to-book ra-tios. However, we stress that these suggestions are basedon past evidence which may not be indicative of futureevents. As you know, economists do reasonably well ex-plaining the past, but are generally a bit shaky when itcomes to predicting the future.

Despite any shortcomings these theories exhibit whenmeasured against historical data, the CAPM and APThave become increasingly important tools for evaluatingcapital investment projects.²⁷

They provide an increas-ingly significant framework for corporate financial ana-lysts who can use them appropriately while understand-ing their limitations.

²⁶However,

Daniel and Titman (1997) find that the market-to-book ratio and market capitalizations of

stocks more accurately predict returns than do the Fama and French factor betas.

²⁷Chapter

11 discusses how firms can use the CAPM and the APTto calculate their costs of equity

capital.

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210Part IIValuing Financial Assets

Key Concepts

Result	6.1:	If securities returns follow a factor model	the variance of r˜can be decomposed into
			i
		(with uncorrelated residuals), portfolios	the sum of K1 terms
		with approximately equal weight on all
			var (˜2˜2˜)
			r) var(F)var(F
		securities have residuals with standard	ii11i22
			. . .2˜)var(˜)
		deviations that are approximately inversely	var(F
			iKKi
		proportional to the square root of the
		number of securities.	Result 6.5:An investment with no firm-specific risk
Result	6.2:	The factor beta of a portfolio on a given	and a factor beta of on the jthfactor in
			ij
		factor is the portfolio-weighted average of	a K-factor model is tracked by a portfolio
		the individual securities’betas on that	with weights of on factor portfolio 1,
			i1
		factor.	on factor portfolio 2,..., on factor
			i2iK
			K
Result	6.3:	Assume that there are Kfactors
			portfolio K,and 1 on the risk-free
		uncorrelated with each other and that the	ij
			j 1
		returns of securities i and j are respectively
			asset. The expected return of this tracking
		described by the factor models
			portfolio is therefore
		˜˜˜. . .
		r FF	...
		iii11i22	r
			fi11i22iKK
		˜˜
		F
		iKKi	where , , ... denote the risk premiums
			12K
		˜ F˜F˜. . .
		r
		jij11j22	of the factor portfolios and ris therisk-free
			f
		˜˜
		F	return.
		jKKj
			Result 6.6:An arbitrage opportunity exists for all
		Then the covariance between r˜and r˜is
		ij	investments with no firm-specific risk
		˜)var(F˜)	unless
		var(F
		iji1j11i2j22
		. . .˜)	. . .
		var(F
		iKjKK	r˜ r
			ifi11i22iKK
Result	6.4:	When Kfactors are uncorrelated with each
			where , ..., applies to all investments
			1
		other and security i is described by the
			with no firm-specific risk.
		factor model
		r ˜ F˜F˜. . .
		iii11i22
		˜˜
		F
		iKKi

Key Terms

arbitrage opportunity176		gross domestic product (GDP)184
arbitrage pricing theory (APT)	176	macroeconomic variables176
common factors176		market model177
diversifiable risk178		multifactor model183
factor analysis184		one-factor model177
factor betas (factor sensitivities)	176	nondiversifiable risk178
factor models176		pure factor portfolios195
factor portfolios184		R-squared178
factor risk176		systematic (market) risk178
firm-specific components176		unsystematic (nonmarket) risk178
firm-specific risk176		well-diversified portfolios178

Exercises

6.1.	Prove that the portfolio-weighted average of a	portfolio and the factor divided by the variance of
	stock’s sensitivity to a particular factor is the same	the factor if the factors are uncorrelated with each
	as the covariance between the return of the	other. Do this with the following steps: (1) Write

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Chapter 6

Factor Models and the Arbitrage Pricing Theory

211

	out the factor equation for the portfolio by	the “zero-beta rate” implied by the factor equations
6.2.6.3.6.4.6.5.6.6.	multiplying the factor equations of the individualstocks by the portfolio weights and adding. (2)Group terms that multiply the same factor.(3)Replace the factor betas of the individual stockreturns by the covariance of the stock return withthe factor divided by the variance of the factor.(4)Show that the portfolio-weighted average of thecovariances that multiply each factor is the portfolio return’s covariance with the factor. The rest is easy. What is the minimum number of factors needed toexplain the expected returns of a group of 10 securities if the securities returns have no firm-specific risk? Why? Consider the following two-factor model for the returns of three stocks. Assume that the factors andepsilons have means of zero. Also, assume the factors have variances of .01 and are uncorrelatedwith each other. ˜r .136F˜F˜˜ 4 A12A ˜r .152F˜F˜˜ 2 B12B ˜r .075F˜F˜˜ 1 C12C If var(˜) .01 var(˜) 0.4 var(˜) .02, ABC what are the variances of the returns of the three stocks, as well as the covariances and correlationsbetween them? What are the expected returns of the three stocks inexercise 6.3? Write out the factor betas, factor equations, and expected returns of the following portfolios:(1)Aportfolio of the three stocks in exercise 6.3 with $20,000 invested in stock A, $20,000 invested in stock B, and $10,000 invested in stock C. (2)Aportfolio consisting of the portfolio formed in part (1) of this exercise and a $3,000 short position in stock C of exercise 6.3. How much should be invested in each of the stocksin exercise 6.3 to design two portfolios? The firstportfolio has the following attributes: factor 1 beta 1 factor 2 beta 0	for the three stocks in exercise 6.3. This is the expected return of a portfolio with factor betas of zero. 6.7.Two stocks, Uni and Due, have returns that follow the one factor model: ˜ .112F˜˜ r uniuni ˜ .175˜˜ rF duedue How much should be invested in each of the two securities to design a portfolio that has a factor beta of 3? What is the expected return of this portfolio, assuming that the factors and epsilons have means of zero? 6.8.Describe how you might design a portfolio of the 40 largest stocks that mimic the S&P500. Why might you prefer to do this instead of investing in all 500 of the S&P500 stocks? 6.9.Prove that (1)rin equation (6.3), iif assuming the CAPM holds. To do this, take expected values of both sides of this equation and match up the values with those of the equation for the CAPM’s securities market line. 6.10.Compute the firm-specific variance and firm- specific standard deviation of a portfolio that minimizes the firm-specific variance of a portfolio of 20 stocks. The first 10 stocks have firm-specific variances of .10. The second 10 stocks have firm- specific variances of .05. 6.11.Find the weights of the two pure factor portfolios constructed from the following three securities: r .06˜F˜ 2F2 112 r .053˜1˜ FF 212 r .04˜F˜ 3F0 312 Then write out the factor equations for the two pure factor portfolios, and determine their risk premiums. Assume a risk-free rate that is implied by the factor equations and no arbitrage. 6.12.Assume the factor model in exercise 6.11 applies again. If there exists an additional asset with the following factor equation ˜F˜ r .081F0 412
	The second portfolio has the attributes: factor 1 beta 0 factor 2 beta 1 Compute the expected returns of these two portfolios. Then compute the risk premiums ofthese two portfolios assuming the risk-free rate is	does an arbitrage opportunity exist? If so, describe how you would take advantage of it. 6.13Use the information provided in Example 6.10 to determine the coordinates of the intersection of the solid and dotted red lines in Exhibit 6.7.

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212Part IIValuing Financial Assets

References and Additional Readings

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Brown, Stephen J., and Mark I. Weinstein. “ANew

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Chen, Nai-fu; Richard Roll; and Stephen A. Ross.

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1996a), pp. 55–84.

———. “The CAPM Is Wanted: Dead or Alive.” Journal

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Grinblatt, Mark, and Sheridan Titman. “Factor Pricing in

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12 (Dec. 1983), pp. 497–507.

———. “Approximate Factor Structures: Interpretations

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Strategy, Second Edition

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Factor Models and the Arbitrage Pricing Theory

213

Gultekin, N. Bulent, and Richard J. Rogalski.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

CHAPTER
7
	Pricing	Derivatives

Learning Objectives

After reading this chapter, you should be able to:

1.Explain what a derivative is and how basic derivatives, like futures, forwards,

options, and swaps, work.

2.Use the binomial model to construct a derivative’s tracking portfolio and

understand its importance in valuing the derivative.

3.Understand how to form arbitrage portfolios if a derivative’s market price differs

from its model price.

4.Use risk-neutral valuation methods to solve for the no-arbitrage prices of any

derivative in a binomial framework and understand why risk-neutral solutions to

the valuation problems of derivatives are identical to solutions based on tracking

portfolios.

In 1994, a group of derivatives valuation experts from the fixed-income proprietary

trading desk of Salomon Brothers, Inc., formed a hedge fund known as Long-Term

Capital Management, L.P. (LTCM). The trading strategy of this fund was to invest in

derivative securities that were undervalued relative to their tracking portfolios and to

short the tracking portfolio. The fund also shorted securities that were overvalued

relative to their tracking portfolios and invested in the tracking portfolios. While the

misvaluations on a security-by-security basis were generally small, LTCM was able

to earn enormous returns on its capital in some years by leveraging its capital; that

is, the fund borrowed money to scale up the size of its positions in the financial

markets. LTCM’s executives felt that the enormous use of leverage was relatively

prudent given that the market values of the derivatives and their tracking portfolios

were so highly correlated. However, after the default by Russia on its sovereign debt

in August 1998, the correlations between the returns of the tracking portfolios and

the derivatives in LTCM’s portfolio declined substantially. As a consequence, to meet

margin calls on its leverage financing, LTCM had to liquidate many of its positions

at huge losses relative to the size of its capital base. (At one point, the fund had lost

about 90% of its capital.) In September 1998, the Federal Reserve stepped in and

called a meeting of a consortium of banks that were investors and lenders to LTCM.

214

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Strategy, Second Edition

Chapter 7

Pricing Derivatives

215

There was substantial concern that the collapse of LTCM would have severe

ramifications on the banks at the meeting as well as on the world financial system.

The banks decided to provide the fund with additional capital to prevent its imminent

collapse. They received control of the fund in exchange for this capital.

In the last few decades, Wall Street has experienced an unprecedented boom that

revolved around the development of new financial products known as derivatives. A

derivativeis a financial instrument whose value today or at some future date is derived

entirely from the value of another asset (or a group of other assets), known as the

underlying asset(assets). Examples of derivatives include interest rate futures con-

tracts; options on futures; mortgage-backed securities; interest rate caps and floors;

swap options; commodity-linked bonds; zero-coupon Treasury strips; and all sorts of

options, contractual arrangements, and bets related to the values of other more primi-

tive securities.

Derivatives have now entered into the public discourse and political debate, as any-

one who has read a newspaper in the 1990s is aware. Long-Term Capital Management,

Metallgesellschaft AG, Orange County, Gibson Greetings, Procter and Gamble, and

Barings Bank, to name a few, received substantial unwanted publicity because of deriv-

ative positions that went sour.

Despite these “scandals,” derivatives remain popular among investors and corpo-

rations because they represent low transaction cost vehicles for managing risk and for

betting on the price movements of various securities. For example, Wall Street investors

shorted the Nikkei and Topix futures contracts to hedge their purchases of some under-

valued Japanese warrants over the last two decades. Savings banks, which finance long-

term loans with short-term deposits, enter into interest rate swaps to reduce the inter-

est rate risk in their balance sheets. Corporations such as Disney, with substantial

revenues from Japan, enter into currency swaps and currency forward rate agreements

to eliminate currency risk. Pension fund managers, with large exposure to stock price

movements, buy put options on stock indexes to place a floor on their possible losses.

Mutual funds that have accumulated large profits may short stock index futures con-

tracts in order to effectively realize capital gains in an appreciating stock market with-

out selling stock, thereby avoiding unnecessary taxes. The list of profitable uses of

derivatives is extensive.

The most important use of derivatives is a risk-reduction technique known as hedg-

ing,¹which requires a sound understanding of how to value derivatives. Derivatives

hedging is based on the notion that the change in the value of a derivative position can

offset changes in the value of the underlying asset. However, it is impossible to under-

stand how a derivative’s value changes without first understanding what the proper

value is.

Some derivatives are straightforward to value and others are complicated. Every

year on Wall Street, hundreds of millions of dollars are made and lost by speculators

willing to test their skills and their models by placing bets on what they believe are

misvalued derivatives.

The major breakthrough in the valuation of derivatives started with a simple call

option. At about the same time that options began trading on an organized exchange

(in 1973 at the Chicago Board Options Exchange), two finance professors at the Mass-

achusetts Institute of Technology, Fischer Black and Myron Scholes, came out with a

1

Hedging is covered in depth in Part VI.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

216Part IIValuing Financial Assets

formula, commonly known as the Black-Scholes formula, that related the price of a

call option to the price of the stock to which the option applies. This formula was the

start of a revolution for both academics and practitioners in their approach to finance.

It generated a Nobel prize in economics for Myron Scholes and Robert Merton (who

provided many key insights in option pricing theory). Fischer Black passed away before

the prize was awarded.

The Black-Scholes Model that led to the development of the formula is now part

of a family of valuation models, most of which are more sophisticated than the Black-

Scholes Model. This family, known as derivatives valuation models, is based on the

now familiar principle of no arbitrage. No-arbitrage valuation principles can determine

the fair market price of almost any derivative. Knowledge of these principles is com-

mon in the financial services industry. Therefore, it is often possible to tailor a secu-

rity that is ideally suited to a corporation’s needs and to identify a set of investors who

will buy that security. Investment banks and other financial intermediaries that design

the security largely concern themselves with what design will be attractive to corpora-

tions and investors. These intermediaries do not worry about the risks of taking bets

opposite to those of their customers when they “make markets” in these derivatives

because they can determine, to a high degree of precision, what the derivative is worth

and know how to hedge their risks.²

The analysis of derivatives in this chapter begins by exploring some of the deriv-

atives encountered by the corporate financial manager and the sophisticated investor,

followed by an examination of valuation principles that apply to all derivatives.³