Encyclopedia of Finance Part 5 pot

Chapter 7 TREASURY INFLATION-INDEXED SECURITIES QUENTIN C. CHU, The University of Memphis, USA DEBORAH N. PITTMAN, Rhodes College, USA Abstract In January 1997, the U.S. Treasury began to issue inflation-indexed securities (TIIS). The new Treas- ury security protects investors from inflation by link- ing the principal and coupon payments to the Consumer Price Index (CPI). This paper discusses the background of issuing TIIS and reviews their unique characteristics. Keywords: treasury inflation-indexed securities; consumer price index; real interest rate; inflation risk premium; phantom income; reference CPI; dutch auction; competitive bidders; noncompeti- tive bidders; bid-to-cover ratio; Series-I bonds. Eleven issues of Treasury inflation-indexed se- curities (TIIS) have been traded in the U.S. market as of December 2003. Inflation-indexed securities are intended to protect investors from inflation by preserving purchasing power. By linking value to the Consumer Price Index (CPI), TIIS provide investors with a ‘‘real’’ rate of return. This security can be viewed as one of the safest financial assets due to its minimal exposure to default risk and uncertain inflation. The fundamental notion behind inflation pro- tection is to preserve the purchasing power of money. Today, inflation protection may be accom- plished by linking investment principal to some form of a price index, such as the Consumer Price Index (CPI) in the United States, Canada, the United Kingdom, and Iceland; the Wholesale Price Index (WPI) in Finland, Brazil, and Argen- tina; and equities and gold in France. In essence, investors purchasing inflation- indexed securities are storing a basket of goods for future consumption. Fifteen countries, includ- ing the United States have issued inflation-indexed securities, starting from the 1940s. 1 Some of the countries had extremely high inflation, such as Mexico and Brazil (114.8 percent and 69.2 percent in the year prior to the introduction of inflation- indexed securities), and others had moderate infla- tion like Sweden and New Zealand (4.4 percent and 2.8 percent). The United Kingdom. has the largest and oldest market for inflation-indexed securities. As of 1997, there were £55 billion index-linked gilts outstand- ing, constituting about 20 percent of all govern- ment bonds in the United Kingdom. The United States is the most recent country to issue inflation- indexed securities to the public. The treasury an- nounced its intention to issue inflation-indexed bonds on May 16, 1996. The first U.S. Treasury inflation-indexed securities were $7 billion of 10-year notes issued in January 1997. There are many motivations for the issue of inflation-indexed securities. First, governments can reduce public financing costs through reducing the interest paid on public debt by the amount of an inflation risk premium. Rates on Treasury securities are usually taken to represent the nom- inal risk-free rate, which consists of the real rate plus expected inflation and an inflation risk pre- mium. By linking value to the price index, infla- tion-indexed securities provide investors with a real rate of interest. This return is guaranteed, whatever the course of inflation. When there is no risk of inflation, the inflation risk premium is re- duced, if not eliminated completely. Benninga and Protopapadakis (1983) revised the Fisher equation to incorporate an inflation risk premium. Second, the issue of inflation-indexed securities is an indication of a government’s intention to fight inflation. A government can keep inflation low through its fiscal and monetary policies. According to the Employment Act of 1946, one of the four primary goals of the U.S. federal government is to stabilize prices through a low-inflation rate. Inflation-indexed securities provide a way for the public to evaluate the government’s performance in controlling inflation. For a constant level of expected inflation, the wider the yield spread between nominal and real bonds, the higher the inflation risk premium, and presumably lower the public’s confidence in the monetary authorities. Moreover, a government promises investors a real rate of return through the issue of inflation- indexed securities. Any loss of purchasing power due to inflation, which investors experience during the investment period, will be offset by inflation- adjusted coupon payments and principal. In an environment with high inflation, the government’s borrowing costs will be high. Reducing borrowing costs provides an incentive for a government to control inflation. The willingness of the govern- ment to bear this risk shows its determination to fight inflation. Inflation-indexed securities also provide a dir- ect measure of expected real interest rates that may help policymakers make economic decisions. According to economic theory, most savings, con- sumption, and investment decisions depend criti- cally on the expected real rate of interest, the interest rate one earns after adjusting the nominal interest rate for the expected rate of in- flation. Real interest rates measure the real growth rate of the economy and the supply and demand for capital in the market. Before the trading of inflation-indexed secur- ities, there was no security in the United States, which was offering coupon and principal payments linked to inflation, and therefore enabling meas- urement of the expected real rate. Empirical studies testing the relationship between expected real rates and other macroeconomic variables have relied instead on indirect measures of the expected real rate such as ex post real rates estimated by sub- tracting actual inflation from realized nominal holding-period returns (Pennachi, 1991). Infla- tion-indexed securities permit the direct study of the real interest rate. Wilcox (1998) includes this as one benefit, which has motivated the Treasury to issue these new securities. Finally, inflation-indexed securities offer an al- ternative financial vehicle for portfolio manage- ment. Since the returns on nominal bonds are fixed in nominal terms, they provide no hedge against uncertain inflation. Kaul (1987) and Chu et al. (1995) have documented a negative correl- ation between equity returns and inflation in the United Kingdom as in the case of investors in equity markets, who suffer during periods of un- expected high inflation. Inflation-indexed secur- ities, by linking returns to the movement of a price index, provide a hedge for investors who have a low-risk tolerance for unexpected inflation. Investors most averse to inflation will purchase inflation-indexed securities, and those less sensitive to inflation will purchase the riskier nominal bonds. The design of the U.S. inflation-indexed secur- ities underwent considerable discussion in deter- mining the linking price index, the cash flow structure, the optimal length of maturity, the auc- tion mechanism, and the amount of issuance. TIIS are auctioned through the Dutch auction method used by other Treasury securities. Participants submit bids in terms of real yields. The highest accepted yield is used to price the newly issued TIIS for all participants (Roll, 1996). 360 ENCYCLOPEDIA OF FINANCE Both principal and coupon payments of TIIS are linked to the monthly nonseasonally adjusted U.S. City Average-All Items Consumer Price Index for All Urban Consumers (CPI-U). The Bureau of Labor Statistics compiles and publishes the CPI independently of the Treasury. The CPI-U is an- nounced monthly. Inflation-indexed securities pro- vide a guarantee to investors that at maturity investors will receive the inflation-adjusted amount or the par value whichever is greater. The coupon payments and the lump-sum payment at maturity are adjusted according to inflation rates. With a fixed coupon rate, the adjustment to a nominal coupon payment is accomplished by multiplying the principal value by one plus the inflation rate between the issuance date and the coupon payment date. Inflation-indexed securities set a floor (par value), an implicit put option, guaranteeing the bond’s value will not fall below its face value if the United States experiences cumulative deflation during the entire life of the TIIS, which is a highly unlikely event. TIIS are eligible for stripping into their principal and interest components in the Treasury’s Separate Trading of Registered Interest and Principal of Securities Program. Since March 1999, the U.S. Treasury Department has allowed all TIIS interest components with the same maturity date to be interchangeable (fungible). Fungibility is designed to improve the liquidity of stripped interest com- ponents of TIIS, and hence increase demand for the underlying inflation-indexed securities. Other Treasury securities are strippable as well. Since first issue in 1997, TIIS have constituted only a small portion of total Treasury securities issuance. At the end of 2002, the market capitaliza- tion of the TIIS was $140 billion, while the total Treasury market capitalization was $3.1 trillion. There are only 11 issues of TIIS outstanding, with original maturities running from 5 to 30 years. The issuance of TIIS was increased from two to three auctions of 10-year TIIS per year, along with a statement from the U.S. Treasury that they actively intend to promote trading in the 10-year note. Lim- ited issuance prevents full coverage for various in- vestment horizons and constrains trading volume in the new security. TIIS have not been closely fol- lowed by financial analysts, nor well understood by the investment public. Since the inception of the TIIS in 1997, actual inflation has been very low by historical standards, and there has not been strong interest in hedging against inflation. Although the Federal Reserve remains concerned about potential inflation, higher inflation levels have not materialized. In more re- cent years, the government has been retiring Treas- ury debt due to government surpluses, which makes significant new issues of TIIS less likely. One disadvantage of TIIS is the potential for tax liability on phantom income. Although the secur- ities are exempt from state and local taxes, they are subject to federal taxation. Positive accrued infla- tion compensation, if any, is reportable income, even though the inflation-adjusted principal will not be received until maturity. Some taxable inves- tors may thus hesitate to invest in TIIS, while others with nontaxable accounts such as retirement accounts might find this market attractive. Conse- quently, investor tax brackets may affect decisions about including TIIS in a portfolio. The emergence of pension funds specializing in TIIS should attract more individual investment in the form of IRA and 401(k) savings, although these investors are more likely to buy and hold. One feature of the TIIS that impedes its use as a perfect measure of the ex ante real rate is the CPI indexing procedure. There is a three-month lag in the CPI indexing system for TIIS. Figure 7.1 indi- cates how the reference CPI is calculated on May 15, 2000. The reference CPI for May 1, 2000, is the CPI-U for the third-previous calendar month, i.e. the announced CPI for February 2000. The Bureau of Labor Statistics surveys price information for the February CPI between January 15 and Febru- ary 15, and then announces the February CPI on March 17, 2000. The reference CPI for any other day of May is calculated by linear interpolation between the CPIs of February and March (the CPI for March becomes available on April 14, 2000). Once the March CPI is announced, the TREASURY INFLATION-INDEXED SECURITIES 361 reference CPI for any day in May 2000 is known. The reference CPI for May 15, 2000 can be calcu- lated according to the following formula: RCPI May15 ¼ CPI Feb þ (14=31) CPI March À CPI Feb ðÞ ¼ 169:7 þ (14=31)(171:1 À169:7) ¼ 170:33226, where RCPI represents the reference CPI for a particular day. 2 The principal value of TIIS on any particular day is determined by multiplying the face value at the issuance by an applicable index ratio. The index ratio is defined as the reference CPI applic- able to the calculation date divided by the refer- ence CPI applicable to the original issuing date. Table 7.1 shows the percentage holdings of TIIS for competitive bidders, noncompetitive bidders, the Federal Reserve, and foreign official institu- tions. The total dollar amount tendered by com- petitive bidders is 2.24 times the total dollar amount accepted. The bid-to-cover ratio of 2.24 indicates the intensity of demand for the TIIS. The first TIIS was issued in January 1997, which offered a real coupon rate of 3.375 percent and 10 years to maturity. The first maturity of TIIS occurred on July 15, 2002. There are eight 10-year TIIS and three 30-year TIIS currently outstanding. Maturities range from 2007 to 2032. Ten-year TIIS original issuances are scheduled in July each year, with a reopening in October and the following January. Each issue has a unique CUSIP number for identification purposes, which is also used in the case of reopening. All 11 issues have been reopened at least once after the original issue date. The average annual return on the 10-year TIIS, since inception in 1997, was 7.5 percent, com- pared to a return on the 10-year nominal Treasury of 8.9 percent. The comparable annual volatility has been 6.1 percent for the TIIS compared to 8.2 percent for the Treasury. Issue size varies from $5 billion to $8 billion. For all 11 issues, the amounts tendered by the public have been consist- ently higher than offering amounts. The average daily trading volume of the TIIS was $ 2 billion, compared to $300 billion for the Treasury market. Jan. Feb. March April May June Survey Period Feb. CPI announced on March 17, 2000 March CPI announced on April 14, 2000 Feb. CPI linked to May 1, 2000 March CPI linked to June 1, 2000 Figure 7.1. Calculation of reference CPI. This figure illustrates the lag effect in indexing the CPI. Due to CPI-U reporting procedures, the reference CPI for May 1, 2000, is linked to the February CPI-U, and the reference CPI for June 1, 2000, is linked to the March CPI-U. Table 7.1. TIIS distribution among investment groupsThe numbers in this table represent auction results of TIIS between October 1998 and July 2001. Information on new issuance and reopening are summarized for the 11 auctions held during this period of time. Amounts are in millions of dollars. Tendered Accepted Competitive 153,446 98.13 68,410 95.90 Noncompetitive 601 0.38 601 0.84 Federal Reserve 2,202 1.41 2,202 3.09 Foreign Official Institutions 125 0.08 125 0.18 Total 156,374 100.00 a 71,338 100.00 a, b a Numbers are in percentage. b Does not add to 100.00 percent because of rounding. 362 ENCYCLOPEDIA OF FINANCE The U.S. Treasury also issues Series-I Bonds, usually called I-Bonds, whose values are linked to the CPI as well. Unlike TIIS, I-Bonds are designed to target individual investors. The motivation for such a security is to encourage public savings. Investors pay the face value of I-Bonds at the time of purchase. The return on I-Bonds consists of two separate parts: a fixed rate of return, and a variable inflation rate. As inflation rates evolve over time, the value of I-Bonds also varies. Values will be adjusted monthly, while interest is com- pounded every six months. Interest payments are paid when the bond is cashed. As in the TIIS, there is an implicit put option impounded in I-Bonds that protects investors from deflation. There are differences between I-Bonds and TIIS. First, I-Bonds are designed for individual investors with long-term commitments. Although investors can cash an I-Bond any time 6 months after issu- ance, there is a 3-month interest penalty if the bond is cashed within the first 5 years. TIIS, on the other hand, can be traded freely without penalty. The real rates of return on I-Bonds and TIIS are different. The Treasury announces the fixed rates on I-Bonds every 6 months, along with the rate of inflation. Both the fixed rate and the inflation rate remain effective for only 6 months until the next announcement date. The real coupon rate on a TIIS, however, is determined through an auction mechanism involving all market participants on the original issue date. TIIS principal is linked to the daily reference CPI, and its value can be adjusted daily instead of monthly as in the case of I-Bonds. The tax treatment of I-Bonds and TIIS is also different. While there is phantom income tax on TIIS, federal income taxes can be deferred for up to 30 years for I-Bonds. If there is early redemp- tion, taxes are levied at the time I-Bonds are cashed. Investors can purchase I-Bonds through retirement accounts, but there is a limit on the amount one can purchase. An investor can pur- chase up to $30,000 worth of I-Bonds each calen- dar year, a limit that is not affected by the purchase of other bond series. NOTES 1. According to the date of introduction of inflation-in- dexed securities, these countries are Finland, France, Sweden, Israel, Iceland, Brazil, Chile, Colombia, Argentina, the United Kingdom, Australia, Mexico, Canada,NewZealand,andtheUnitedStates. 2. The U.S. Treasury posts the reference CPI for the following month around the 15th of each month on its web site at http:==www.publicdebt.treas.gov. REFERENCES Benninga, S. and Protopapadakis, A. (1983). ‘‘Nominal and real interest rates under uncertainty: The Fisher theorem and the term structure.’’ The Journal of Political Economy, 91(5):856–867. Chu, Q.C., Lee, C. F., and Pittman, D.N. (1995). ‘‘On the inflation risk premium.’’ Journal of Business, Finance, and Accounting, 22(6):881–892. Kaul, G. (1987). ‘‘Stock returns and inflation: The role of the monetary sector.’’ Journal of Financial Eco- nomics, 18(2): 253–276. Pennachi, G. G. (1991). ‘‘Identifying the dynamics of real interest rates and inflation: Evidence using sur- vey data.’’ Review of Financial Studies, 4(1):53–86. Roll, R. (1996). ‘‘U.S. Treasury inflation-indexed bonds: The design of a new security.’’ The Journal of Fixed Income 6(3):9–28. Wilcox, D. W. (1998). ‘‘The introduction of indexed government debt in the United States.’’ The Journal of Economic Perspectives, 12(1):219–227. TREASURY INFLATION-INDEXED SECURITIES 363 Chapter 8 ASSET PRICING MODELS WAYNE E. FERSON, Boston College, USA Abstract The asset pricing models of financial economics de- scribe the prices and expected rates of return of securities based on arbitrage or equilibrium theories. These models are reviewed from an empirical per- spective, emphasizing the relationships among the various models. Keywords: financial assets; arbitrage; portfolio op- timization; stochastic discount factor; beta pricing model; intertemporal marginal rate of substitution; systematic risk; Capital Asset Pricing Model; con- sumption; risk aversion; habit persistence; durable goods; mean variance efficiency; factor models; arbitrage pricing model Asset pricing models describe the prices or expected rates of return of financial assets, which are claims traded in financial markets. Examples of financial assets are common stocks, bonds, op- tions, and futures contracts. The asset pricing models of financial economics are based on two central concepts. The first is the ‘‘no arbitrage principle,’’ which states that market forces tend to align the prices of financial assets so as to elim- inate arbitrage opportunities. An arbitrage oppor- tunity arises if assets can be combined in a portfolio with zero cost, no chance of a loss, and a positive probability of gain. Arbitrage opportun- ities tend to be eliminated in financial markets because prices adjust as investors attempt to trade to exploit the arbitrage opportunity. For example, if there is an arbitrage opportunity because the price of security A is too low, then traders’ efforts to purchase security A will tend to drive up its price, which will tend to eliminate the arbitrage opportunity. The arbitrage pricing model (APT), (Ross, 1976) is a well-known asset pricing model based on arbitrage principles. The second central concept in asset pricing is ‘‘financial market equilibrium.’’ Investors’ desired holdings of financial assets are derived from an optimization problem. A necessary condition for financial market equilibrium in a market with no frictions is that the first-order conditions of the investor’s optimization problem are satisfied. This requires that investors are indifferent at the margin to small changes in their asset holdings. Equilib- rium asset pricing models follow from the first- order conditions for the investors’ portfolio choice problem, and a market-clearing condition. The market-clearing condition states that the aggregate of investors’ desired asset holdings must equal the aggregate ‘‘market portfolio’’ of securities in supply. Differences among the various asset pricing models arise from differences in their assumptions about investors’ preferences, endowments, produc- tion and information sets, the process governing the arrival ofnewsinthe financialmarkets,and the types of frictions in the markets. Recently, models have been developed that emphasize the role of human imperfections in this process. For a review of this ‘‘behavioral finance’’ perspective, see Barberis and Shleifer (2003). Virtually all asset pricing models are special cases of the fundamental equation: P t ¼ E t {m tþ1 (P tþ1 þ D tþ1 )}, (8:1) where P t is the price of the asset at time t and D tþ1 is the amount of any dividends, interest or other pay- ments received at time t þ 1. The market wide ran- dom variable m tþ1 is the ‘‘stochastic discount factor’’ (SDF). By recursive substitution in Equa- tion (8.1), the future price may be eliminated to express the current price as a function of the future cash flows and SDFs only: P t ¼ E t {S j>0 (P k¼1 , , j m tþk )D tþj }. Prices are obtained by ‘‘discounting’’ the payoffs, or multiplying by SDFs, so that the expected ‘‘present value’’ of the payoff is equal to the price. We say that a SDF ‘‘prices’’ the assets if Equa- tion (8.1) is satisfied. Any particular asset pricing model may be viewed simply as a specification for the stochastic discount factor. The random vari- able m tþ1 is also known as the benchmark pricing variable, equivalent martingale measure, Radon– Nicodym derivative, or intertemporal marginal rate of substitution, depending on the context. The representation in Equation (8.1) goes at least back to Beja (1971), while the term ‘‘stochastic discount factor’’ is usually ascribed to Hansen and Richard (1987). Assuming nonzero prices, Equation (8.1) is equivalent to: E t (m tþ1 R tþ1 À 1) ¼ 0, (8:2) where R tþ1 is the vector of primitive asset gross returns and 1 is an N-vector of ones. The gross return R i,tþ1 is defined as (P i,tþ1 þ D i,tþ1 )=P i,t , where P i,t is the price of the asset i at time t and D i,tþ1 is the payment received at time t þ 1. Em- pirical tests of asset pricing models often work directly with asset returns in Equation (8.2) and the relevant definition of m tþ1 . Without more structure the Equations (8.1,8.2) have no content, because it is always possible to find a random variable m tþ1 for which the equa- tions hold. There will be some m tþ1 that ‘‘works,’’ in this sense, as long as there are no redundant asset returns. For example, take a sample of asset gross returns with a nonsingular covariance matrix and let m tþ1 be :[1 0 (E t {R tþ1 R tþ1 0 }) À1]R tþ1 Substi- tution in to Equation (8.2) shows that this SDF will always ‘‘work’’ in any sample of returns. The ability to construct an SDF as a function of the returns that prices all of the included assets, is essentially equivalent to the ability to construct a minimum-variance efficient portfolio and use in as the ‘‘factor’’ in a beta pricing model, as described below. With the restriction that m tþ1 is a strictly posi- tive random variable, Equation (8.1) becomes equivalent to the no arbitrage principle, which says that all portfolios of assets with payoffs that can never be negative but are positive with positive probability, must have positive prices (Beja, 1971; Rubinstein, 1976; Ross, 1977; Harrison and Kreps, 1979; Hansen and Richard, 1987.) While the no arbitrage principle places restric- tions on m tþ1 , empirical work more typically ex- plores the implications of equilibrium models for the SDF based on investor optimization. A repre- sentative consumer–investor’s optimization implies the Bellman equation: J(W t ,s t )  max E t {U(C t ,:) þJ(W tþ1 ,s tþ1 )}, (8:3) where U(C t ,:) is the utility of consumption expend- itures at time t , and J(.) is the indirect utility of wealth. The notation allows that the direct utility of current consumption expenditures may depend on other variables such as past consumption ex- penditures or the current state variables. The state variables, s tþ1 , are sufficient statistics, given wealth, for the utility of future wealth in an opti- mal consumption–investment plan. Thus, the state variables represent future consumption–invest- ment opportunity risk. The budget constraint is: W tþ1 ¼ (Wt À C t )x 0 R tþ1 , where x is the portfolio weight vector, subject to x 0 1 ¼ 1. If the allocation of resources to consumption and investment assets is optimal, it is not possible to obtain higher utility by changing the allocation. Suppose an investor considers reducing consump- tion at time t to purchase more of (any) asset. The ASSET PRICING MODELS 365 expected utility cost at time t of the foregone con- sumption is the expected product of the marginal utility of consumption expenditures, Uc(C t ,:) > 0 (where a subscript denotes partial derivative), multiplied by the price of the asset, and which is measured in the same units as the consumption expenditures. The expected utility gain of selling the investment asset and consuming the proceeds at time t þ1isE t {(P i,tþ1 þ D i,tþ1 ) J w (W tþ1 ,s tþ1 )}. If the allocation maximizes expected utility, the following must hold: P i,t E t {U c (C t ,:)} ¼ E t {(P i,tþ1 þD i,tþ1 ) J w (W tþ1 ,s tþ1 )} which is equ- valent to Equation (8.1), with m tþ1 ¼ J w (W tþ1 ,s tþ1 ) E t {U c (C t ,:)} : (8:4) The m tþ1 in Equation (8.4) is the ‘‘intertemporal marginal rate of substitution’’ (IMRS) of the con- sumer–investor. Asset pricing models typically focus on the rela- tion of security returns to aggregate quantities. It is therefore necessary to aggregate the first-order conditions of individuals to obtain equilibrium ex- pressions in terms of aggregate quantities. Then, Equation (8.4) may be considered to hold for a representative investor who holds all the securities and consumes the aggregate quantities. Theoretical conditions that justify the use of aggregate quan- tities are discussed by Gorman (1953), Wilson (1968), Rubinstein (1974), and Constantinides (1982), among others. When these conditions fail, investors’ heterogeneity will affect the form of the asset pricing relation. The effects of heterogeneity are examined by Lintner (1965), Brennan and Kraus (1978), Lee et al. (1990), Constantinides and Duffie (1996), and Sarkissian (2003), among others. Typically, empirical work in asset pricing fo- cuses on expressions for expected returns and ex- cess rates of return. The expected excess returns are modeled in relation to the risk factors that create variation in m tþ1 . Consider any asset return R i,tþ1 and a reference asset return, R 0,tþ1 . Define the excess return of asset i, relative to the reference asset as r i,tþ1 ¼ R i,tþ1 À R 0,tþ1 . If Equation (8.2) holds for both assets it implies: E t {m tþ1 r i,tþ1 } ¼ 0 for all i: (8:5) Use the definition of covariance to expand Equation (8.5) into the product of expectations plus the covariance, obtaining: E t {r i,tþ1 } ¼ Cov t (r i,tþ1 ; Àm tþ1 ) E t {m tþ1 } , for all i, (8:6) where Cov t (:;:) is the conditional covariance. Equation (8.6) is a general expression for the expected excess return from which most of the expressions in the literature can be derived. Equation (8.6) implies that the covariance of return with m tþ1 , is a general measure of ‘‘system- atic risk.’’ This risk is systematic in the sense that any fluctuations in the asset return that are uncor- related with fluctuations in the SDF are not ‘‘priced,’’ meaning that these fluctuations do not command a risk premium. For example, in the conditional regression r itþ1 ¼ a it þ b it m tþ1 þ u itþ1 , then Cov t (u itþ1 , m tþ1 ) ¼ 0. Only the part of the variance in a risky asset return that is correlated with the SDF is priced as risk. Equation (8.6) displays that a security will earn a positive risk premium if its return is negatively correlated with the SDF. When the SDF is an aggregate IMRS, negative correlation means that the asset is likely to return more than expected when the marginal utility in the future period is low, and less than expected when the marginal utility and the value of the payoffs, is high. For a given expected payoff, the more negative the cov- ariance of the asset’s payoffs with the IMRS, the less desirable the distribution of the random re- turn, the lower the value of the asset and the larger the expected compensation for holding the asset given the lower price. 8.1. The Capital Asset Pricing Model One of the first equilibrium asset pricing models was the Capital Asset Pricing Model (CAPM), 366 ENCYCLOPEDIA OF FINANCE developed by Sharpe (1964), Lintner (1965), and Mossin (1966). The CAPM remains one of the foundations of financial economics, and a huge number of theoretical papers refine the assump- tions and provide derivations of the CAPM. The CAPM states that expected asset returns are given by a linear function of the assets’ ‘‘betas,’’ which are their regression coefficients against the market portfolio. Let R mt denote the gross return for the market portfolio of all assets in the economy. Then, according to the CAPM, E(R itþ1 ) ¼ d 0 þ d 1 b i ,(8:7) where b i ¼ Cov(R i , R m )=Var(R m ): In Equation (8.7), d 0 ¼ E(R 0tþ1 ), where the return R 0tþ1 is referred to as a ‘‘zero-beta asset’’ to R mtþ1 because the condition Cov(R 0tþ1 , R mtþ1 ) ¼ 0. To derive the CAPM, it is simplest to assume that the investor’s objective function in Equa- tion (8.3) is quadratic, so that J(W tþ1 , S tþ1 ) ¼ V{E t (R ptþ1 ), Var t (R ptþ1 )} where R ptþ1 is the inves- tor’s optimal portfolio. The function V(.,.) is increasing in its first argument and decreasing in the second if investors are risk averse. In this case, the SDF of Equation (8.4) specializes as: m tþ1 ¼ a t þ b t R ptþ1 . In equilibrium, the representative agent must hold the market portfolio, so R ptþ1 ¼ R mtþ1 . Equation (8.7) then follows from Equation (8.6), with this substitution. 8.2. Consumption-based Asset Pricing Models Consumption models may be derived from Equa- tion (8.4) by exploiting the envelope condition, U c (:) ¼ J w (:), which states that the marginal utility of current consumption must be equal to the mar- ginal utility of current wealth, if the consumer has optimized the tradeoff between the amount con- sumed and the amount invested. Breeden (1979) derived a consumption-based asset pricing model in continuous time, assuming that the preferences are time-additive. The utility function for the lifetime stream of consumption is S t b t U(C t ), where b is a time preference parameter and U(.) is increasing and concave in current con- sumption, C t . Breeden’s model is a linearization of Equation (8.1), which follows from the assump- tion that asset values and consumption follow diffusion processes (Bhattacharya, 1981; Gross- man and Shiller, 1982). A discrete-time version follows Lucas (1978), assuming a power utility function: U(C) ¼ [C 1Àa À 1]=(1 À a), (8:8) where a > 0 is the concavity parameter of the period utility function. This function displays constant relative risk aversion equal to a. ‘‘Relative risk aver- sion’’ in consumption is defined as: Cu 00 (C)=u 0 (C). Absolute risk aversion is defined as: u 00 (C)=u 0 (C). Ferson (1983) studied a consumption-based asset pricing model with constant absolute risk aversion. Using Equation (8.8) and the envelope condi- tion, the IMRS in Equation (8.4) becomes: m tþ1 ¼ b(C tþ1 =C t ) Àa : (8:9) A large body of literature in the 1980s tested the pricing Equation (8.1) with the SDF given by the consumption model (Equation (8.9)). See, for ex- ample, Hansen and Singleton (1982, 1983), Ferson (1983), and Ferson and Merrick (1987). More recent work generalizes the consumption- based model to allow for ‘‘nonseparabilities’’ in the U c (C t ,:) function in Equation (8.4), as may be implied by the durability of consumer goods, habit persistence in the preferences for consump- tion, nonseparability of preferences across states of nature, and other refinements. Singleton (1990), Ferson (1995), and Cochrane (2001) review this literature; Sarkissian (2003) provides a recent em- pirical example with references. The rest of this section provides a brief historical overview of empirical work on nonseparable-consumption models. Dunn and Singleton (1986) and Eichenbaum et al. (1988) developed consumption models with durable goods. Durability introduces nonsepar- ability over time, since the actual consumption at a given date depends on the consumer’s previous expenditures. The consumer optimizes over the ASSET PRICING MODELS 367 current expenditures C t , accounting for the fact that durable goods purchased today increase con- sumption at future dates, and thereby lower future marginal utilities. Thus, U c (C t ,:) in Equation (8.4) depends on expenditures prior to date t. Another form of time nonseparability arises if the utility function exhibits ‘‘habit persistence.’’ Habit persistence means that consumption at two points in time are complements. For example, the utility of current consumption may be evaluated relative to what was consumed in the past, so the previous standard of living influences the utility derived from current consumption. Such models are derived by Ryder and Heal (1973), Becker and Murphy (1988), Sundaresan (1989), Constan- tinides (1990), and Campbell and Cochrane (1999), among others. Ferson and Constantinides (1991) model both durability and habit persistence in consumption expenditures. They show that the two combine as opposing effects. In an example based on the utility function of Equation (8.8), and where the ‘‘mem- ory’’ is truncated at a single-lag, the derived utility of expenditures is: U(C t ,:) ¼ (1 À a) À1 S t b t (C t þ bC tÀ1 ) 1Àa ,(8:10) where the coefficient b is positive and measures the rate of depreciation if the good is durable and there is no habit persistence. Habit persistence implies that the lagged expenditures enter with a negative effect (b < 0). Empirical evidence on similar habit models is provided by Heaton (1993) and Braun et al. (1993), who find evidence for habit in inter- national consumption and returns data. Consumption expenditure data are highly sea- sonal, and Ferson and Harvey (1992) argue that the Commerce Department’s X-11 seasonal adjust- ment program may induce spurious time series behavior in the seasonally adjusted consumption data that most empirical studies have used. Using data that are not adjusted, they find strong evidence for a seasonal habit model. Abel (1990) studied a form of habit persistence in which the consumer evaluates current consump- tion relative to the aggregate consumption in the previous period, and which the consumer takes as exogenous. The idea is that people care about ‘‘keeping up with the Joneses.’’ Campbell and Cochrane (1999) developed another model in which the habit stock is taken as exogenous (or ‘‘external’’) by the consumer. The habit stock in this case is modeled as a highly persistent weighted average of past aggregate consumptions. This ap- proach results in a simpler and more tractable model, since the consumer’s optimization does not have to take account of the effects of current decisions on the future habit stock In addition, by modeling the habit stock as an exogenous time series process, Campbell and Cochranes’ model provides more degrees of freedom to match asset market data. Epstein and Zin (1989, 1991) consider a class of recursive preferences that can be written as: J t ¼ F(C t , CEQ t (J tþ1 )). CEQ t (:) is a time t ‘‘cer- tainty equivalent’’ for the future lifetime utility J tþ1 . The function F(:, CEQ t (:)) generalizes the usual expected utility function and may be nontime-separable. They derive a special case of the recursive preference model in which the prefer- ences are: J t ¼ (1 À b)C p t þ b E t (J 1Àa tþ1 ) p=(1Àa) hi 1=p : (8:11) They show that the IMRS for a representative agent becomes (when p 6¼ 0, 1 À a 6¼ 0): m tþ1 ¼ [ b (C tþ1 =C t ) pÀ1 ] (1Àa)=p {R m,tþ1 } ((1ÀaÀp)=p) : (8:12) The coefficient of relative risk aversion for time- less consumption gambles is a and the elasticity of substitution for deterministic consumption is (1 Àp) À1 .Ifa ¼ 1 À p, the model reduces to the time-separable power utility model. If a ¼ 1, the log utility model of Rubinstein (1976) is obtained. Campbell (1993) shows that the Epstein–Zin model can be transformed to an empirically tractable model without consumption data. He used a line- arization of the budget constraint that makes it 368 ENCYCLOPEDIA OF FINANCE [...]... (1988) ‘‘A theory of rational addiction.’’ Journal of Political Economy, 96: 6 75 700 Beja, A (1971) ‘‘The structure of the cost of capital under uncertainty.’’ Review of Economic Studies, 4: 359 –369 Berk, J (19 95) ‘‘A critique of size-related anomalies.’’ Review of Financial Studies, 8: 2 75 286 Bhattacharya, S (1981) ‘‘Notes on multiperiod valuation and the pricing of options.’’ Journal of Finance, 36:... Premiums.’’ Journal of Political Economy, 99: 3 85 4 15 374 ENCYCLOPEDIA OF FINANCE Ferson, W.E and Harvey, C.R (1992) ‘‘Seasonality and consumption based Asset Pricing Models.’’ Journal of Finance, 47: 51 1 55 2 Ferson, W.E and Jagannathan, R (1996) ‘‘Econometric evaluation of Asset Pricing Models,’’ in G.S Maddala and C.R Rao (eds.) Handbook of Statistics: volume 14: Statistical Methods in Finance Amsterdam,... equilibrium APT: application of a new test methodology.’’ Journal of Financial Economics, 21: 255 –290 Constantinides, G.M (1982) ‘‘Intertemporal Asset Pricing with heterogeneous consumers and without demand Aggregation.’’ Journal of Business, 55 : 253 –267 Constantinides, G.M (1990) ‘‘Habit formation: a resolution of the equity premium puzzle.’’ Journal of Political Economy, 98: 51 9 54 3 373 Constantinides,... Journal of Political Economy, 99: 263–286 Fama, E.F and French, K.R (1993) ‘‘Common risk factors in the returns on stocks and bonds.’’ Journal of Financial Economics, 33: 3 56 Fama, E.F and French, K.R (1996) ‘‘Multifactor explanations of asset pricing anomalies.’’ Journal of Finance, 51 : 55 –87 Ferson, W.E (1983) ‘‘Expectations of real interest rates and aggregate consumption: empirical tests.’’ Journal of. .. Journal of Political Economy, 107: 2 05 251 Chamberlain, G (1983) ‘‘Funds, factors and diversification in arbitrage pricing models.’’ Econometrica, 51 : 13 05 1324 Chamberlain, G and Rothschild, M (1983) ‘‘Arbitrage, factor structure and mean variance analysis on large asset markets.’’ Econometrica, 51 : 1281–1304 Chen, N.F (1983) ‘‘Some empirical tests of the theory of arbitrage pricing.’’ Journal of Finance, ... heterogenous consummers.’’ Journal of Political Economy, 104: 219–240 Cox, J.C., Ingersoll, J.E., and Ross, S.A (19 85) ‘‘A theory of the term structure of interest rates.’’ Econometrica, 53 : 3 85 408 Cragg, J.G and Malkiel, B.G (1982) Expectations and the Structure of Share Prices Chicago: University of Chicago Press Dunn, K.B and Singleton, K.J (1986) ‘‘Modelling the term structure of interest rates under non-separable... under non-separable utility and durability of goods.’’ Journal of Financial Economics, 17: 27 55 Dybvig, P.H (1983) ‘‘An explicit bound on individual assets’ deviations from APT pricing in a finite economy.’’ Journal of Financial Economics, 12: 483–496 Dybvig, P and Ingersoll, J (1982) ‘‘Mean variance theory in complete markets.’’ Journal of Business, 55 : 233– 252 Eichenbaum, M.S., Hansen, L.P., and Singleton,... information.’’ Journal of Financial Economics, 10: 1 95 210 Hansen, L P and Richard, S F (1987) ‘‘The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models.’’ Econometrica, 55 : 58 7–613 Hansen, L.P and Singleton, K.J (1982) ‘‘Generalized instrumental variables estimation of nonlinear rational expectations models.’’ Econometrica, 50 : 1269–12 85 Hansen, L.P and... ‘‘Mimicking portfolios and exact arbitrage pricing.’’ Journal of Finance, 42: 1–10 Jobson, J.D and Korkie, R (1982) ‘‘Potential performance and tests of portfolio efficiency.’’ Journal of Financial Economics, 10: 433–466 Kandel, S and Stambaugh, R.F (1989) ‘‘A mean variance framework for tests of asset pricing models.’’ Review of Financial Studies, 2: 1 25 156 Lee, C.F., Wu, C., and Wei, K.C.J (1990) ‘‘The heterogeneous... theory and implications.’’ Journal of Financial and Quantitative Analysis, 25: 361–376 Lehmann, B.N and Modest, D.M (1988) ‘‘The empirical foundations of the arbitrage pricing theory.’’ Journal of Financial Economics, 21: 213– 254 Lintner, J (19 65) .‘‘The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets.’’ Review of Economics and Statistics, 47: 13–37 . structure of the cost of capital under uncertainty.’’ Review of Economic Studies, 4: 359 –369. Berk, J. (19 95) . ‘‘A critique of size-related anomalies.’’ Review of Financial Studies, 8: 2 75 286. Bhattacharya,. Economics, 33: 3 56 . Fama, E.F. and French, K.R. (1996). ‘‘Multifact or ex- planations of asset pricing anomalies.’’ Journal of Finance, 51 : 55 –87. Ferson, W.E. (1983). ‘‘Expectations of real interest. (CAPM), 366 ENCYCLOPEDIA OF FINANCE developed by Sharpe (1964), Lintner (19 65) , and Mossin (1966). The CAPM remains one of the foundations of financial economics, and a huge number of theoretical