Ch.. Asset Pricing Theory (7383S) Juha Joenväärä University of Oulu March 04 Abstract This chapter introduces the modern asset pricing theory based on the stochastic discount factor approach. The main idea is that asset prices should be equal to discounted expected payo. I start reviewing the main concepts related to expected utility and risk aversion. Indeed, the expected utility provides a convenient way to rank risky investments between each other. Next, I turn on state pricing. I introduce a basic state price rule to price assets. It bases heavily to so called primitive securities, which can be used to price other assets. I also present conditions when state prices exists. These conditions include the absence of arbitrage and the law of one price. Finally, I focus on stochastic discount approach to price nancial assets. First, I solve the maximization problem of representative investor. Using the rst order conditions of the problem, one can nd the stochastic discount factor that can be used to price asset. In the modern nance, the systemic risk of an asset is captured by the covariance with the stochastic discount factor. Indeed, assets with a positive (negative) covariance with the stochastic discount factor has low (high) expected return.
Asset Pricing Theory The expected utility: Provides a convenient way to rank risky investments between each other. Risk aversion, risk premium and certainty equivalence. State pricing. Bases heavily to so called primitive securities, which can be used to price other assets. State prices exists if conditions such as the absence of arbitrage and the law of one price holds The stochastic discount factor approach to asset pricing. Solving a representative investor s maximization problem can be obtained stochastic discount factor. Using the rst order conditions, one can nd the stochastic discount factor that can be used to price asset. In the modern nance, the systemic risk of an asset is captured by the covariance with the stochastic discount factor. Indeed, assets with a positive (negative) covariance with the stochastic discount factor has low (high) expected return. Does Consumption based Asset Pricing work in practice? Derivation of CCAPM Empirical evidence Equity Premium Puzzle Potential solutions for Equity Premium Puzzle
State-pricing. Utility theory and risk aversion Time and Risk dimensions Prefer for smooth consumption stream. It s often assumed that individuals have a desire for a smooth stream of consumption. Plan A: Investor consumes units in year and 0 units in year two. Plan B: Investor consumes 6 units both years. Most investors choose plan B. Time and Risk dimensions: Time dimension: It s di cult to alter standard of living from period to period in response to a highly variable consumption stream. Risk dimension: Individuals would prefer to have the same standard of living in the next period no matter what events will take place. The role of nancial markets: If investor generally prefer a smooth consumption stream, we can assume that they would be better o if they could diversify away some of their consumption risk. Financial market o ers instruments that allow for this kind of diversi cation. Time dimension:. Borrow money to consume today. Pay back to debt in a future period from your consumption in that period.. Invest today in a asset that o ers you a payo in a future period. Risk dimension:. An asset that has a variable payo can also diversify away some the risks in the investor s consumption ows. (Think this issue in SDF framework; What kind of assets have a positive correlation with SDF i.e., marginal rate of substitution? How about their expected return, low or high?) 3
Example. Assume that investor follows Plan with current consumption (c 0 ) is 9 units. Plan : Asset Asset c 0 c ;s x ;s x ;s State. (s=) c 0 = 9 c ; = 0 x ; = x ; = 4 State. (s=) c 0 = 9 c ; = x ; = x ; = 3 State 3. (s=3) c 0 = 9 c ;3 = 8 x ;3 = 3 x ;3 = State 4 (s=4) c 0 = 9 c ;4 = 4 x ;4 = 4 x ;4 = She does not know for sure the state of nature of next period. However, there are two assets having state-contingent payo s presented in the table above. Which one she chooses if she does not have to pay anything to obtain of on these assets? Both assets have a same expected payo equaling to.5 units Of course, generally investors prefer Asset, because then her consumption will be less variable. Thus, investors can diversify consumption risks away. However, we do not have yet tools to value these assets, i.e., to assign a price and expected return for them! Expected Utility Theory: Provides a criterion for comparing and ranking di erent investments.. Specify preferences (utility function) for an investor.. Obtain a single number that serves a ranking between di erent investments. The utility function de ned in a this way is a von Neumann-Morgenstern (VNM) utility function. The utilities, V () ; that are assigned to di erent gambles, portfolios, or consumption levels can then used to rank these objects. The ranking will be in relation to investors preferences that are described by the utility function U () of which we are are taking expectation over. Speci cally, if we have a consumption plan fc ;;:::; c ;s g de ned over S mutually exclusive states of nature, the utility that the investor would obtain in each of these states is V (c ) = E [U (c )] = SX S U (c ;S ) : s= A VNM utility-maximizing investor will then choose the consumption plan that maximizes his expected utility. We will later maximize an investor s expected utility in order to nd SDF! 4
Risk aversion How can we understand risk aversion? We say that an investor is risk averse if she would not accept a fair gamble, where a fair gamble is de ned as one that has an expected value of zero. Consider a gamble where an investor can either gain an amount h with probability / or lose an amount h with probability /. Because this is a gamble with a zero expected payo, a risk averse investor with a level of personal wealth W would not participate it. This implies that the expected utility that he gets fro the two possible outcomes of the gamble is less that the utility he receives from keeping his personal wealth safe in his hand U (W ) > U (W + h) + U (W h) = E (U) : To capture this implication, we make two assumptions on the utility function U (W ) :. U 0 (W ) > 0: The rst derivative of the utility function is positive. This implies that investor prefers more to less. His utility function increase whenever his wealth, consumption or whatever we use as an argument in the utility function increases.. U 00 (W ) < 0: The second derivative of the utility function is negative. The utility increases at a decreasing speed. In other words, the utility that the investor receives from a xed increase in his wealth gets smaller the larger his current wealth is (the utility increase for a poor man is higher than the utility increase for a rich man when wealth increase by, say, 00 units). This captures risk aversion in the investor s preferences, and it also implies that risk averse investor would rather have a smooth level of consumption rather than a variable one around a xed mean. The two assumptions imply that the utility function is strictly increasing and strictly concave. Absolute risk aversion: R A (w) U 00 (W ) U 0 (W ) : Measures risk aversion for a given level of wealth. For example, when the wealth level increases, will more or less be invested in risky assets? Often we assume decreasing ARA. 5
Relative risk aversion: R R (w) W U 00 (W ) U 0 (W ) : How does the relative amount of the wealth invested in risk assets change when the wealth level increases? Often we assume constant RRA. Risk Premium and certainty equivalence Consider a risk averse investor, with current wealth W, evaluating an uncertain risky project payo x t+ : Then for any distribution function F W, we obtain using Jensen s inequality U (W + E [x t+ ]) > E [U (W + x t+ )] : This implies that uncertain payo is available for sale, a risk averse investor will only be willing to buy it at a price less that its expected payo. Certainty equivalent: The maximal certain sum of money a person is willing to pay to acquire an uncertain opportunity. Risk premium: The di erence between the certainty equivalent and expected value of the payo. Think these de nitions in a SDF framework, p t = E (m t+ x t+ ) : What p t really means in this equation? How risk premium is related to m t+? We try to understand later! 6
. State pricing Next, we construct a simple one-period state-pricing model. The nancial asset is characterized by its price (p) at time 0 and its payo (x) at time time. The payo of an asset can be viewed as the sum of the ex-dividend price and the end of the period and the dividend x t+ = p t+ + d t+ : This means that if you pay a stock today for a price of p t ; the payo at the end of the period is the ex-dividend price plus the dividend that you will receive. In a one-period model we assume that the whole value of the asset is paid as a dividend to the investor at the end of the period. I drop out time notations t in order to make notations easier; However remember that price p is always today s price and payo x is always tomorrow s payo. To incorporate uncertainty (risk) in the analysis, we assume that there are S di erent and mutually exclusive states of nature that can occur during the period. The concepts of risk the arises because the investor, making his investment decisions at time 0, does not know with certainty the state of nature that will occur. Each state of nature is associated with probability S, which sum up to one SX S = : s= The state-contingent payo of an asset, that is indexed by i, can then be presented by a payo vector 3 x ;i x ;i x i = 6 7 4. 5 Let s assume that we have the earlier example with equal state probabilities x S;i s Asset Asset x ;s x ;s State = =4 x ; = x ; = 4 State = =4 x ; = x ; = 3 State 3 3 = =4 x ;3 = 3 x ;3 = State 4 4 = =4 x ;4 = 4 x ;4 = 7
If there are N assets, each will have a payo vector x i ; i = f; ; : : : ; Ng, then the payo vectors can be stacked in a payo matrix x that contains all the state-contingent payo s of the individual assets. This matrix will have a dimension of S N (S rows and N columns). 3 x ; x ;N 6 x = 4... 7.. 5 : x S; x S;N Let p represent the beginning period prices of theses N assets (an N vector) 3 6 p = 4 Consider an asset that provides a payo if and only if state s occurs, and a payo of 0 in all other states Denote the price of such an asset by q s : p. p N 7 5 : The payo vector, denoted by e s, of such an asset will be 3 3 x. 0. e s = x s = : 6 7 6 7 4. 5 4. 5 0 x S If there exists such assets for each state of economy, i.e., there exists e s for s = f; ; : : : ; Sg, we can stack the payo vectors of these assets in an S S identity matrix 3 0 0 0 0 0.. e =.. 0.. 0. : 6 7 4.. 0 0 5 0 0 0 The state price vector of these assets is denoted by q (an S vector) 3 6 q = 4 These assets, whose payo is denoted by e s ; are called primitive securities, Arrow-Debreu securities, pure securities, state securities and state contingent claims in the literature 8 q. q S 7 5 :
These primitive securities are the building blocks of modern asset pricing theory. Note that the state price, q s ; gives the current price of one unit of payo that we get in state s: Since individual assets are described by their state-contingent payo s, and if we know the prices, given by q s ; of these state-contingent payo s, we can use the state prices to assign a price to any other asset on the market. Prices of primitive securities are the state price vector ( the superscript T denotes a matrix transpose) 6 4 6 4 p. p N p. p N p = x T q; x = e 3 0 0 7 6 5 = 4 0... 0 0 0 3 3 7 5 = 6 4 q. q S 7 5 3 7 6 5 4 q. q S 3 7 5, The pricing formula is simply given by (the superscript T denotes a matrix transpose) Example : p = x T q; There are three states of the economy, i.e., S = 3: The payo vectors of the assets are x = (8:5; 0; 4) T, x = (8:5; 8; 0) T and x 3 = (7; 8; 6) T : There are also three primitive assets, one for each state of the economy, with the state price vector q = (0:8; 0:45; 0:30) T : The prices of these three assets are set using p = x T q 4 4 p p p 3 p p p 3 3 5 = 3 5 = = 4 4 4 x ; x ; x ;3 x ; x ; x ;3 x 3; x 3; x 3;3 x ; x ; x 3; x ; x ; x 3; x ;3 x ;3 x 3;3 8:5 0 4 8:5 8 0 7 8 6 3 5 3 5 4 T 3 5 4 4 q q q 3 q q q 3 0:8 0:45 0:30 3 3 5 3 5 5 = 4 7:3 8:3 6:66 3 5 9
In actual nancial markets, these primitive securities do not traded directly. However, under some conditions, we can infer their prices from the assets that are traded. Finding the state price vector q from prices and payo s of the traded assets is called a reverse decomposition problem. The condition that must hold if we are to nd a unique state price vector q by using the observed prices and payo s of individual assets are the following Theorem : Market Completeness For unique state prices for each state to exists, the payo matrix x must be of rank S, rank(x) = S: This implies that there are as many linearly independent assets as there are states of nature. Then, the market is said to be complete, the matrix x is invertible, and state prices can be recovered by calculating q = x T p: () Example 3: Consider the same setting as in previous example, but there is also the forth asset with a payo vector x 4 = (4; 6; ) T : In the line of the law of one price, the price of Asset 4 is p 4 = 4:0: The stacked payo matrix is now and the rank of x is 3, rank(x) = 3. x = 4 8:5 8:5 7 4 0 8 8 6 4 0 7 This implies that there are 3 linearly independent securities on the market. Since there are three states of nature, the market is complete. In order to take inverse of the payo matrix, we need to have a square matrix (as many rows as columns). We can take out one of the assets to get rid of one column and check if the remaining matrix still has a rank of three. Taking out, for example Asset 3, we are left with a matrix 8:5 8:5 3 4 x = 4 0 8 6 5 4 0 that still has a full rank. 3 5 0
Then, states can be recovered using formula. 4 q q q 3 q = x T p 3 3 8:5 0 4 5 = 4 8:5 8 0 5 4 6 3 0:8 = 4 0:45 5 0:30 4 7:3 8:3 4:0 3 5 Using the prices for the individual assets and their payo s, we arrive at the same state prices that we de ned in the previous example.
.3 The Law of One Price The law of one price states that two assets having the same state contingent payo must have the same price. Formally,.let p (x ) be the current price of an asset with a payo vector x ; then the Law of One Price implies that we can nd a linear pricing rule. De nition : The Law of one Price If the Law of One Price holds, then we have a linear pricing rule in the form of p ( x + x ) = p (x ) + p (x ) ; where is the amount invested in asset and is the amount invested in asset. The Law of One Price states that the price of a portfolio consisting individual securities, must be given by the prices of these individual securities. It s not possible to repackage two asset into one portfolio and sell the portfolio at a higher price that what is implied by the prices on the individual assets. A violation of this law would give rise to an immediate kind of arbitrage pro t, as you could sell the expensive version and buy the cheap version of the same portfolio. Theorem. State prices exists if and only if the Law of One Price holds. This implication goes for both directions: The law of One Price implies the existence of state prices. The existence of state prices implies that the Law of One Price holds.
.4 Absence of arbitrage Absence of arbitrage says that you cannot get for free a portfolio that might pay o positively. De nition : Arbitrage An arbitrage is a portfolio satisfying one of the following conditions (i) p < 0 and x > 0 (ii) p 6 0 and x > 0 with Prob x > 0 > 0: Condition (i) suggests that we get something today, without having any negative out ows from the portfolio in the future. Condition (ii) implies that we can construct a portfolio today that costs us nothing, but that has a positive probability of producing a positive cash ow in the future. Indeed, you should not be able to construct a portfolio that will certainly not cost you anything, but that might pay of positively. This de nition is di erent from the colloquial use of word arbitrage. Most people use arbitrage to mean a violation of the law of one price - a riskless way of buying something cheap and selling it for a higher price. Arbitrage here might pay o, but they again might not. Absence of arbitrage implies the following theorem. Theorem 3: The positivity of state prices If there is no arbitrage opportunities on the market, then the state prices will be strictly positive. This implies both direction: The positivity of state prices implies absence of arbitrage. The absence of arbitrage implies the positivity of state prices To understand this, assume that the state prices are strictly positive (q s > 0 for all states s) : Recall that both of the arbitrage conditions require that the payo s from portfolio must be either zero or positive. The prices of all portfolios are give by the sum of the products between the state prices and the payo s that portfolio produces. This implies that products between state prices and payo s must be either zero or positive, and thus, the price of the portfolio cannot be negative. If the portfolio produces a positive payo, its price today must be positive if the state price is positive. 3
Thus, we cannot have any arbitrage if state prices are positive x T {z } all elements either positive or zero q {z} all elements strictly positive = p {z} never negative The Law of One Price and Absence of Arbitrage are two distinct concepts. The absence of arbitrage implies that the law of one price holds. If the law of one price did not hold, then we would have an arbitrage opportunity. On the other hand, the law of the one price does not imply absence of arbitrage opportunities. There can be arbitrage opportunities even if the law of one price holds. Example 3: Suppose there are two assets and two possible states of nature. The payo s from the two assets are x = (0; ) T and x = (; ) T with prices p = 0:9 and p = :6: Because payo s are not linearly dependent (Rank(x) = ), the market is complete. Suppose that the law of one price holds, so that price of any portfolio is simply given by the prices of the individual assets, p = p + p = (0:9) + (:6) : Consider now a new portfolio that shorts two units of asset and buys one unit of asset two. That is, = and = : Because the law of one price holds, the price of such a portfolio p = (0:9) + (:6) = 0:: The state contingent payo s of the portfolio are x = 0 Thus, the portfolio has a negative price today, and never produces a negative payo and it has a positive probability for a positive payo. Hence, we clearly have arbitrage even thought the law of one price holds. We know from the theorem above that positive state prices guarantee the absence of arbitrage opportunities. Hence, because there is an arbitrage opportunity in this example, we cannot have positive state prices. Using the formula that allows us to recover state prices in a complete market, we get : : q = x T p 0 0:9 q = :6 = 0: 0:9 Thus, the rst state price is negative, implying that there are arbitrage opportunities. 4
.5 Summary and the most important ideas We have studied how to assign asset prices given the existence of state prices. In addition, we have examined the conditions under which these state prices, and consequently, an asset pricing formula exists. The existence of state prices: If Law of one price holds, there will always exists state prices. If the stronger condition, the absence of arbitrage holds, not only do these state prices exists, but there will be strictly positive state prices. If markets are complete, there exists a unique state vector (only one). I expect that you understand the most important concepts after this section: Expected utility and risk aversion. Basics of state pricing: The role of primitive securities i.e., Arrow-Debreu assets. Markets completeness. How the pricing formula works. Meaning of the absence of arbitrage. Meaning of the law of one price. The conditions about existence of state prices. Next, we turn on the economic determinants of state prices. 5
3 Stochastic discount factor approach How state prices are determined? What are economic sources determining why a unit of payo is more valuable in some states of nature and less valuable in some other states? Because every asset can be priced with these state prices, the answer to the question posed above should also help us to understand why some assets have lower prices than some other assets. 3. Maximization problem of representative investor Assume that there exists a representative investor. A representative investor is somewhat abstract concept. When posing existence of such an investor, we assume that actions of all investors can be aggregated in such a way that we need to study the actions of one investor who is aggregate of the individual investors. Rubinstein (974) shows that there are several technical conditions under such an aggregation is possible to do. We assume that these conditions are ful lled. The representative investor much choose how much to consume today and how much to consume in the s possible di erent states of nature tomorrow. The maximization problem of the representative investor: max E [U (C)] = U(c 0 ) + c 0 ;c ;:::c s SX s U (c s ) ; s= {z } Objective function SX Subject to c 0 + q s c s = W; s= {z } Budget Constraint where subjective time-preference factor, S probability of state s occurring, q S is the state price for state s, W is the investor s current wealth. The investor chooses his consumption levels today (t) and tomorrow (t + ) in such a way that his expected utility is maximized. The budget constraint states that the price of the consumption pattern (c 0 ; c ; : : : c s ) must equal his current wealth (W ). 6
This means that he cannot today allocate more wealth to consumption than he owns. However, the budget constrain makes sure that he consumes all the wealth during either the rst period (t) or second period (t + ) : We further assume that the price of one unit of consumptions today is unit, q 0 = : We can solve this constrained optimization problem using Lagrangian method: L : U(c 0 ) + SX s U (c s ) + W c 0 + s= First order conditions for c 0 @L : U 0 (c 0 ) = 0 @c 0, U 0 (c 0 ) =!! SX q s c s s= First order conditions for c s @L : s U 0 (c s ) q s = 0 @c s, s U 0 (c s ) = q s Using FOCs : s U 0 (c s ) = U 0 (c 0 )q s U 0 (c s ) State Price s : q s = s ; s = ; : : : ; S: U 0 (c 0 ) 7
3. Economic determinants of state prices To understand what economic determinants drives state prices (q S ) i.e., the unit value of future payo in a given state s, let s have look are the state-price equation. is the investor s subjective time-preference factor. U 0 (c s ) q s = s ; s = ; : : : ; S: () U 0 (c 0 ) The lower is the value of ; the more value the investor values consumption today relative tomorrow. Low makes all the state prices smaller suggesting that one would rather consume wealth today that tomorrow. One is not willing to pay much today for a unit of consumption (payo ) that one will receive tomorrow.. s is the probability that state s is occurring. One is not willing to pay much for a unit of consumption (payo ) tomorrow if there is a small probability that one will get the payo. 3. The third quantity, U 0 (c s) U 0 (c 0 ; is the most important important for asset pricing purposes. ) U 0 (c s ) is the marginal utility of the consumer/investor. The important economic questions is to understand U 0 (c s ) and the associated payo for a some speci c state of nature s: Since investors are assumed to be risk averse: (a) The rst derivative of utility function is U 0 (c s ) is always positive. (b) The second derivative of U 00 (c s ) is always negative. This implies that marginal utility U 0 (c s ) is high when consumption level c s is low. Hence, state prices q S will be high in states of aggregate consumptions is low. State prices q S are high exactly in states of nature where investors value extra unit of payo most. One is willing to pay much for an assets that gives you high payo s in states where you are hungry and where one might otherwise be starving. Indeed, that kind of assets diversify away some of the risks in your consumption ows and increase one s utility. The denominator, U 0 (c 0 ), is the same for all states, thus one can only concentrate on the nominator U 0 (c s ): 8
3.3 From state prices to stochastic discount factors The equation () involves probabilities for each state of nature s. Thus, it s di cult to compare state prices with each other on a relative basis. Hence, it s hard to compare the relative goodness and badness of states with each other. To get rid of the probability measure s in equation () ; one can divide state prices q s with the probability of state happening q s = U 0 (c s ) s U 0 (c 0 ) m s: where m s is called to the stochastic discount factor! The name - stochastic discount factor - is was used by Hansen and Richard (987). Other names for SDF that appear in literature are state price de ator, state price density, pricing kernel or change of probability measure. 3.4 Asset Pricing with stochastic discount factors Following pricing framework for state prices: p i = = = SX q s x s;i s= SX s= s q s s x s;i SX s m s x s;i s= p i = E [mx i ] Example 4: Recall earlier example where an investors has to choose between to di erent assets. The table shows investor s consumption plan. Plan : Asset Asset c 0 c ;s x ;s x ;s State. c 0 = 9 c ; = 0 x ; = x ; = 4 State. c 0 = 9 c ; = x ; = x ; = 3 State 3. c 0 = 9 c ;3 = 8 x ;3 = 3 x ;3 = State 4 c 0 = 9 c ;4 = 4 x ;4 = 4 x ;4 = 9
We agreed earlier that Asset looks like interesting, since it delivers high payo when investor s consumption is low. Then, however, we didn t had tools to verify our intuition. Using the stochastic discount factor we price these assets. m s = U 0 (c s ) U 0 (c 0 ) ; However, we need to make couple of assumptions about investor preferences.. Investor has a logarithmic utility function U(c) = ln (c) with the rst derivative U 0 (c) = c :. Investor s subjective time discount parameter is 3. There are equal state probabilities = 0:85: s = 4 Using these facts, one can calculate the value of the stochastic discount factor for each state of nature m s = c s = c 0 ; s = ; ; 3; 4: c 0 c s m = 0:85 9 0 = 0:383 m = 0:638 m 3 = 0:956 m 4 = :93 Using formula p i = E [mx i ] = one can calculate the prices of Assets and. SX s m s x s;i ; s= 0
The price for Asset : The price for Asset : p = 4 m x ; + 4 m x ; + 4 m 3 x ;3 + 4 m 4 x ;4 = 3:044 p = 4 m x ; + 4 m x ; + 4 m 3 x ;3 + 4 m 4 x ;4 = :87: Thus, investor has a hunger for Asset. 3.5 Link SDF to returns and risk-free rate Asset pricing s sledgehammer m is ready to use! Indeed, the price of any nancial asset is given by the expectation of the product between the asset s payo and stochastic discount factor Note that I take back t notations for time! p t = E [m t+ x t+ ] : (3) We usually divide the payo x t+ by the price p t to obtain a gross return: R t+ x t+ p t ; where R t+ is the gross return on the asset ( + r t+ ) : We can think of a return as a payo that has price equal to one: if we pay one euro today, the return is how many euros or units or consumption we get tomorrow. Thus, returns obey p t = E m x t+, p t p t = E (m t+ R t+ ) : Returns are commonly used in empirical work, since they are typically stationary (the means, variances and autocovariances are independent of time) over time: they do not have trends.
If there is no uncertainty, we can express returns = E (m t+ R t+ ) = E (m t+ ) R f (4) where R f is the gross risk-free rate. The risk-free rate is related to the discount factor by R f = =E(m t+ ): (5) Since R f is typically greater than one, the payo x t+ sells at a discount. 3.6 Understanding Risk Corrections Using the de nitions of covariance cov(m t+ ; x t+ ) = E(m t+ x t+ ) E(m t+ )E(x t+ ); we can write the pricing equation p t = E(m t+ x t+ ) as p t = E(m t+ )E(x t+ ) + cov(m t+ ; x t+ ): Substituting the risk-free rate equation (5), we obtain where p t = E(x t+) R f + cov(m t+ ; x t+ ); (6) the rst term is the standard discounted present-value formula, giving the asset s price in a risk-neutral world (where consumption is constant or utility is linear) the second term is risk adjustment. Using expected returns, we obtain risk premia or expected return E(R t+ ) R f = R f cov(m t+ ; R t+ ) (7) that is higher for assets having a large negative covariance with the discount factor. This is an extremely important result! I hope that you understand the meaning of this after taking this course! This helps us to understand why some assets have higher expected returns than others. Indeed, equation (7) shows that assets with returns that have a positive covariance with a stochastic discount factor have a low risk premium. The intuition should be clear from a previous example. The SDF m t+ can be theoretically understood as a function of investor marginal utilities U 0 (c):
The SDF obtains high (low) values in states where marginal utility is high (low). An asset with a positive (negative) covariance with the SDF yields high (low) returns in states with high (low) values of the SDF. If covariance term cov(m t+ ; R t+ ) in equation (7) is positive, this means that such an asset have a lower expected return that a risk-free asset. This due to fact that such an asset o ers insurance against bad states of nature. On the other hand, if covariance term cov(m t+ ; R t+ ) in equation (7) is negative, this means that such an asset have a higher expected return that a risk-free asset. A higher negative covariance implies higher risk and expected return, thus investors are also willing to include such an asset into their portfolios. Intuitively, that kind of assets increase the volatility of an investor s consumption stream. Idiosyncratic (asset-speci c) risk does not a ect prices. One important implication of equations (6) and (7) is that the variance of the asset s returns, or it s payo s, does not a ect prices or expected returns. Only the covariance with the SDF matters, since if cov(m t+ ; R t+ ) = 0 ) E(R t+ ) R f = 0 This prediction is true no matter how large is the variance of the asset return (V ar R t+ ): Indeed, one of main principles of modern nance is that only the systematic risk of an asset s payo should be priced or rewarded higher expected risk. In the modern nance, the systemic risk of an asset is captured by the covariance with the stochastic discount factor. Example 5: Consider the previous example. The cross risk-free rate is i.e., the risk-free rate is.86%. R f = E [m] = = SP s m s s= 0:97 = :086 3
Using prices calculated earlier, one can nd that the expected returns are Asset E [r ] = E [x ] p = 7:9% Asset E [r ] = E [x ] p = 37:6%: Thus, expected return on Asset is negative, where as the expected return on Asset is positive. How can be understand the these somewhat puzzling return patterns? First, note that covariances of assets returns with the stochastic discount factor Cov [R ; m] = E [R m] E [R ] E [m] Cov [R ; m] = 0:06 = ( 7:9) (0:97) Cov [R ; m] = 0:3377: Asset has a positive covariance with the SDF. Hence, it delivers high returns in bad states of economy when these high returns are needed. Therefore investors are willing to take a sort of insurance against bad state of economy. Asset has a negative covariance with the SDF. Thus, Asset increases the volatility of consumption stream. It contains risk, therefore investors require a high premium in order to invest in this asset. Using equation (7), we can also retrieve the risk premiums on the assets. E(R) R f = R f cov(m; R) Taking Asset as an example, one can easily nd that R f cov(m; R) = :086 ( 0:3377) = 0:3474 being equal to E(R) R f = :376 :086 = 0:3474: Finally, consider Asset 3 with payo s (:64; 3:498; 0:97; :907) This asset has a volatile payo stream. Hence, it must be a risky asset? Or must it? One calculate its price p 3 = :4307: 4
The expected return on the Asset 3 is (the expected payo is.5) E [r 3 ] = E [x 3] p 3 = :5 :4307 which turns out to be equal to risk-free rate of return! E [r 3 ] = r f: = :85%; This can be understood by calculating covariance between asset return and the SDF Cov [R 3 ; m] = E [R 3 m] E [R 3 ] E [m] = E [R f ] E [m] = E [m] E [m] = = 0 Thus, Asset 3 is uncorrelated with the SDF. Even though asset has a variable payo stream, this variation is not connected to systemic risk - all the risk involved in the asset s payo is unsystematic (idiosyncratic). Therefore, one could conclude that asset risk-free and earns a risk-free rate of return. What one can learn these example! One could erroneously conclude that Asset is better that Asset, due to fact that Asset has a negative return and Asset has a positive return. However, such claims are completely unfounded if one simply looks at raw returns without resorting asset pricing model. Asset provides investors with high returns simply because it is a bad asset - its payo contains more risk than the payo s of Asset. In market equilibrium, the fact that a company is good (excellent management, many investment opportunities, new and promising product lines) is re ected in a high price of the company today. However, expected returns should, on average, only be a compensation for the systematic riskiness of stock s payo s. Furthermore, in market equilibrium, there are no good and bad assets. Prices and expected returns are set in such a way that the badness of a given stock (measure using systematic risk) is exactly balance by higher expected return that it must o er. Thus, investors will, at margin, be indi erent between di erent assets in their portfolio choices when market reaches equilibrium. 5
3.7 Existence of stochastic discount factors There is a close correspondence between the SDF and state prices, one can obtained very similar existence theorems for the SDF. Theorem 4: The fundamental theorem of asset pricing The following conditions on prices p and payo s x are equivalent (i) Absence of arbitrage (ii) Existence of a consistent positive linear pricing rule (positive state prices) (9q >> 0) p = x T q (iii) Some agent with strictly increasing preference U has an optimum Theorem is based on Cox and Ross (976) and Ross (977, 978). It s also know as the fundamental theorem of nancial economics. The points (i) and (ii) familiar form case of state prices. The condition for the existence of any asset pricing model is that state prices are strictly positive and that there are no arbitrage opportunities. The point (iii) implies that if there were in fact arbitrage opportunities, then the representative investor s maximization problem could not react an optimum. Buying more and more consumption in a state that has a negative state price would increase the investor s wealth in nitely, and his expected utility would not be bounded from above. 6
4 Consumption based Asset Pricing Model (CCAPM) When we write the basic model we do not assume p = E (mx) ;. Markets are complete, or there is a representative investor. Asset returns or payo s are normally distributed, or independent of time 3. Two-period investors, quadratic utility, or separable utility 4. Investors have no human capital or labor income 5. The market has reached equilibrium, or individuals have bought all the securities they want to These assumptions come in special cases. But, we do assume that the investor can consider a small marginal investment or disinvestment. The basic pricing equation should hold for any asset (stock, bond, option, real investment opportunity, etc. ), and any monotone and concave utility function. The consumption-based model is, in principle, a complete answer to all asset pricing questions, but works poorly in practice. This observation motivates other asset pricing models. All we need in CCAPM is a functional form for utility, numerical values for the parameters, and a statistical model for the conditional distribution of consumption and payo s. Consider the standard power utility function u 0 (c) = c : Then excess returns should be 0 = E t " ct+ c t R e t+ # : Taking unconditional expectations and applying the covariance decomposition, expected excess returns follow " # E Rt+ e = R f ct+ cov ; Rt+ e : c t 7
4. Empirical Findings Theoretically, the CCAPM appears preferable to the traditional CAPM It takes into account the dynamic nature of portfolio decisions. It integrates the many forms of wealth beyond nancial asset wealth. Consumption should deliver the purest measure of good and bad times as investors consume less when their income prospects are low or if they think future returns will be bad. However, empirically, the original version of the consumption-based model has not been a great success. Hansen and Singleton (98, 983) formulate a consumption-based model in which a representative agent has time-separable power utility of consumption. They reject the model on U.S. data, nding that it cannot simultaneously explain the timevariation of interest rates and the cross-sectional of average returns on stocks and bonds. Wheatley (988) rejects the model based on international data. Mankiew and Shapiro (986) show that the CCAPM performs no better, and in many respects even worse than the CAPM. They regress the average returns of the 464 NYSE stocks that were continuously traded from 959 to 98 on their market betas, on consumption growth betas, and on both betas. They nd that the market betas are more strongly and robustly associated with their cross section of average returns, and that market beta drives out consumption beta in multiple regressions. Breeden, Gibbons, and Lizenberger (989) nd comparable performance of the CAPM and a model that uses a mimicking portfolio for consumption growth as the single factor. Recently, Lettau and Ludvigson (00), Amir and Bansal (004) and Savov (0) among others suggest that Consumption Strikes Back! Savov (0, JF) Asset Pricing with Garbage A new measure of consumption, garbage, is more volatile and more correlated with stocks than the canonical measure, National Income and Product Accounts (NIPA) consumption expenditure. A garbage-based consumption capital asset pricing model matches the U.S. equity premium with relative risk aversion of 7 versus 8 and evades the joint equity premium-risk-free rate puzzle. These results carry through to European data. In a cross-section of size, value, and industry portfolios, garbage growth is priced and drives out NIPA expenditure growth. 8
4. The Basic Equity Premium Puzzle 4.. Equity Premium Puzzle The Hansen-Jagannathan (99) bounds are the characterizations of the discount factors that price a given set of asset returns. Manipulating 0 = E(mR e ), we nd Proof: je(r e )j (R e ) {z } Sharpe ratio 0 = E(mR e ) (m) E(m) : 0 = E(m)E(R e ) + cov(m; R e ) Dividing both sides by E(m) m implies that 0 = E(m)E(R e ) + mr e (m) (R e ) je(r e )j (R e ) (m) E(m) = mr f The highest Sharpe ratio is associated with portfolios lying on the mean-variance e cient frontier. Notice that the slope of the frontier is governed by the volatility of the discount factor. Under the CCAPM it follows that je(r e )j (R e ) = ct+ c t E ct+ c t The postwar U.S. mean value weighted NYSE is about 8% per annum over the T-bill rate, with a standard deviation of about 6%. Thus, the market Sharpe ratio E (R e ) =(R e ) is about 0.5 (8/6) for an annual investment horizon. If there were a constant risk-free rate, E(m) = =R f would nail down E(m). The T-bill rate is not very risky, hence E(m) is not far from the mean of the inverse of the mean T-bill rate (=( + r f )), or about E(m) 0:99. Thus, the facts about the mean return, 8%, and volatility, 6%, imply (m) > 0:5; which means that the volatility of the discount factor must be about 50% of its level in annual data! Per capita consumption growth has standard deviation about % per year; and with log utility that implies ct+ c t = 0:0 = %, which is o by a factor of 50. To match the equity premium we need = 50, which seems a huge level of risk aversion! 9
4.. Correlation Puzzle The H-J bounds take the extreme possibility that consumption and stock returns are perfectly correlated. They are not. The correlation of annual stock returns and nondurable plus services consumption growth in postwar U.S. data is no more than 0.. If we use this information, the calculation becomes (m) E(m) je(r e )j j m;r e j (R e ) = 0:5 = :5: 0: With (m) (c); we now need a risk aversion coe cient of 50! 4..3 Risk-Free Rate Puzzle Average Interest Rates and Subjective Discount Factors: Traditionally, we consider risk aversion numbers = [; 5] or so. What is wrong with = 50 to 50? The most basic piece of evidence for low comes from the relation between consumption growth and interest rates: " # R = E(m ct+ t+) = E f t ; that can be expressed in continuous time, c t r f t = + E t [c] ( + ) t (c) : Real interest rates are typically quite low, about %. With a % mean and % standard deviation of consumption growth, the predicted interest rate rises quickly as we raise. For example, with = 50 and a typical = 0:0 (%), we predict r f = 0:0 + 50 0:0 50 5 0:0 = 0:38 or 38%! Using =, the risk-free rate should be around 5% to 6% per year. The actually observed rate is less than %. To get a reasonable % real interest rate, we have to use a subjective discount factor on negative 37%. That is not impossible such economic model can be speci ed where present values can converge with negative discount rates - but it doesn t seem very reasonable; since people prefer earlier utility. 30
4..4 How Shall We Resolve the Equity Premium and Risk-Free Puzzles? Perhaps investors are much more risk averse than we may have thought. This indeed resolves the equity premium puzzle. But higher risk aversion parameter implies higher risk-free rate. So, higher risk aversion reinforces the risk-free puzzle. Perhaps the stock returns over the last 50 years are good luck rather than an equilibrium compensation for risk. If so, the equity premium will disappear in several decades as more reasonable returns are realized. Perhaps something is deeply wrong with the utility function speci cation and/or the use of aggregate consumption data. Indeed, the CCAPM assumes that agents preferences are time additive von Neumann Morgenstern expected utility representation (e.g., power utility). Standard power utility preferences impose tight restrictions on the relation between the equity premium and the risk-free rate. In power utility, the elasticity of intertemporal substitution (EIS) and the relative risk aversion parameters are reciprocals of each other - economically they should not be tightly linked. EIS is about deterministic consumption paths it measures the willingness to exchange consumption today with consumption tomorrow for a given risk-free rate; whereas risk aversion is about preferences over random variables (lotteries). 3