Chapter 12.2: General asset pricing model Fan Longzhen July, 2003 Consumption-Based Model and Basic Pricing Model ? Basic question to decide for an investor: ? (1) how much to save; ? (2) how much to consume; ? (3)what portfolio of assets to hold. ? Pricing equation come from the first order condition for this decision. Marginal utility and its indicator ? Marginal utility, not consumption, is the fundamental ,measure of how you feel. ? Theory of asset pricing is about how to go from marginal utility to observable indicators. ? Consumption is an indicator of marginal utility. When consumption is low, marginal utility is high. ? A large index may be an indicator. A large index goes down, the wealth of investors goes down, consumption goes down, marginal utility becomes high. Basic Pricing Equation ? Basic pricing problem is to price stream of any uncertainty cash flows; ? Any asset, for example, a stock, with price and dividend next period. The payoff next period is ? What is value of the payoff today for a typical investor. ? We need a model to model typical investor’s utility ( happiness) of consumption ? ? The utility functions may be a power utility form [ ])()(),( 11 ++ += ttttt cUEcUccU β γ γ ? ? = 1 1 1 )( tt ccU 1+t p 1+t d 111 +++ += ttt dpx utility ? Utility function satisfy conditions: ? (1) desire for more consumption, increasing function; ? (2)marginal utility is decreasing; ? (3) impatience for time: β--subjective discount factor; ? (4) aversion for risk:curvature of utility function Consumption and investment decision ? If today's price of the payoff Pt, how much will the typical investor buy and sell? ? Denote by e the original consumption level(if the investor buy none of the asset), and denote by the amount of the asset he choose to buy. The problems is ? The first-order condition for an optimal consumption and portfolio choice ?Or ξ {} ξ ξ β ξ 111 1 .. )]([)( max +++ + += ?= + ttt ttt ttt xec pects cuEcu )1.1(])('[)(' 11 ++ = ttttt xcuEcup β )2.1()]('/)('[ 11 ttttt cuxcuEp ++ = β Pricing formula ? Equation (1.2) is the central pricing formula. ? It relate one endogenous variable, price, to the two other two exogenous variables, consumption and payoff. ? You can interpret consumption in terms of more fundamental, but we stop now. )2.1()]('/)('[ 11 ttttt cuxcuEp ++ = β Stochastic discount factor ? If we define the stochastic discount factor as ? The basic pricing formula can simply expressed as ? Why called discount factor, if no uncertainty, we can express price via the standard present value formula ? is just like 1/(1+Rf)—traditional discount factor. ? It says something deep, all assets use a stochastic factor to adjust risk; ? are also called marginal rate of substitution, pricing kernel, change of measure, or state-price density. t m )(' )(' 1 1 t t t cu cu m + + =β )( 11 ++ = tttt xmEp )1/( ftt Rxp += 1+t m 1+t m Many other names of mt+1 ? Marginal rate of substitute; ? Pricing kernel; ? Change of measure; ? State-price density ? For gross return , the pricing formula becomes ttt pxR / 11 ++ = )(1 11 ++ = ttt RmE Prices, payoff, and notation ? This pricing formula is for any financial asset, stock, bond, option, and future, swap. ? Prices or returns can also be real or nominal. The only difference is we use a real or nominal discount factor. ? If P and X are nominal, we can create real pricing formula as: price p(t) payoff p(t+1) stock p(t) p(t+1)+d(t+1) return 1 R(t+1) price-dividend ratio p(t)/d(t) (p(t+1)/d(t+1)+1)d(t+1)/d（ t） excess return 0 one-period bond p（ t） 1 option c max(S（ T） -1) ] )(' )(' [/ 1 11 + ++ Π ? ? ? ? ? ? ? ? =Π t t t t ttt x cu cu Ep β ] )(' )(' [ 1 1 1 + + + Π Π ? ? ? ? ? ? ? ? = t t t t t tt x cu cu Ep β Class issues in finance ? A simple manipulation of the basic pricing formula can gives a lot of intuition about classic issues in finance—including determinants of the interest rate, risk corrections, idiosyncratic versus syncratic risk, beta pricing model, mean variance- variance frontiers, the slope of mean- variance frontier, time-varying expected returns, and present value relations. Risk-free rate ? A bond pay 1 RMB at t+1, its value today is ? The return of the bond is risk-free rate ? So ? If a risk-free asset is not traded, we can define as “shadow” risk-free rate, sometimes it is also called “zero-beta” rate. ? Use power utility ? By turning of uncertainty, we obtain ? we see three effects right away: ? 1. Real interest rates are high when people are impatient, i.e., when ? is low. If everyone wants to consume now, it takes a high interest rate to convince them to save; ? 2. Real interest rates are high when consumption growth is high. High interest rates lower the level of consumption now, while raising its growth rate from today to tomorrow. ? 3 real interest rate are more sensitive to consumption growth if the power parameter is large. When is large, the utility is highly curved ? 1+= ff rR )( 1+ = ttt mEp )(/1 1+ = ttf mER )(/1 1+tt mE γ? =ccu )(' γ β ? ? ? ? ? ? ? ? = + t t f c c R 1 1 β γ γ Risk-free rate under uncertainty ? When there is uncertainty, the risk-free interest rate is ? We have the same results for risk free rate, but captures precautionary savings. When consumption is more volatile, people with this utility function are more worried low consumption states than they are pleased at high consumption states. ? control two aversions: one: intertemporal substitution; two, risk aversion to varies across state. δ β σ γ γδ ? ++ == ???+= eRr ccEr f t f t tttt f t ;ln ln( 2 )ln( 1 2 2 1 ） 2 σ Risk corrections ? From pricing formula ? We have ?or ? The first term, standard present value formula in the risk-neutral word; the second term, risk adjustment. ? Payoff covaries positively with the discount factor haves a higher value. ? To better understand risk adjustment, substitute m with consumption ? Marginal utility u’? decline as c increases, an asset’s price is lowered if its payoff covariates positively with consumption. )( 11 ++ = tttt xmEp ),cov()()( 1111 ++++ += ttttttt xmxEmEp ),cov(/)( 111 +++ += ttfttt xmRxEp )('/)),('cov(/)( 111 tttfttt cuxcuRxEp +++ += β Risk corrections-continued ? To go further, investor care about the volatility of the consumption, not his portfolio, if he buys more risk asset, his consumption volatility tomorrow is )var(),cov(2)var()var( 2 xxccxc ξξξ ++=+ Using the returns ? Start from basic pricing formula ? We have ?Using ? We get ?Or ? All assets have an expected return equal to the risk-free rate, plus a risk- adjustment. Assets whose returns covary positively with consumption make consumption more volatile, an so must promise a higher expected return to induce investor to hold them. ? Conversely, assets that covary negatively with consumption, such as insurance, can offer a rate lower than risk-free rate. )(1 i mRE= ),cov()()(1 ii RmREmE += )(/1 mER f = )]('[ )),('cov[ )( 1 11 + ++ ?=? t i tt fi cuE Rcu RRE ),cov()( iffi RmRRRE ?=? Idiosyncratic risk does not affect prices ? Volatile asset is risky, however if the payoff is not correlated with the discount factor m, the asset receives no risk correction to its price, if ?Then ? Generally, we can decompose x as ? is the idiosyncratic part that is uncorrelated with discount factor . ? The price of is zero; ? The words “systematic” and “idiosyncratic” are defined differently in different contexts. ? Here, they are uncorrelated with the the discount factor, but they can be correlated with each other. 0),cov( =xm f R Ex p = ε+= m mE mxE x )( )( 2 )()() )( )( ())(( 2 2 xpmxE mE mxE mEmxprojp === ε ε Expected Return-beta representation ? To go further, the expected return can be expressed as ?Or ? This is beta pricing model. ? is interpreted as the price of risk, is quantity of risk in each asset. ?With ? We can express beta in terms of a more concrete variable, consumption growth, rather than marginal utility ) )( )var( ( )var( ),cov( )( mE m m mR RRE i fi ?+= mmi fi RRE λβ , )( += m λ β γ β )/( 1 tt ccm + = )var( )( , c RRE c cci fi ?= += ? ?? γλ λβ To be continued ? Expected returns should increase linearly with their betas on consumption growth itself. ? Price of risk is determined by risk aversion and volatility of consumption. The more risk averse of the people, or the riskier of the environment, an larger expected risk premium the investor required. Mean-Variance Frontier ? Asset pricing theory has focused a lot on the means and variances of asset returns. Interestingly, the set of means and variances of returns is limited. All asset priced by the discount factor m must obey ? To derive it ?Hence )( )( )( )( ifi R mE m RRE σ σ ≤? )()()()()(1 , mRREmEmRE i Rm ii i σσρ+== )( )( )( )( , i Rm fi R mE m RRE i σ σ ρ?= This simple calculation has many interesting and classic implications Mean variance frontier Rf Slope σ(m)/E(m) Ri Idiosyncratic risk σ(R ) E( R) This simple calculation has many interesting and classic implications—to be continued ? 1. The boundary of the mean variance region in which assets can lie is called the mean-variance frontier ? 2. All returns on the frontier are perfectly correlated with the discount factor. The return in the upper part of the frontier are perfectly negatively correlated with the discount factor, and hence positively correlated with consumption. They are maximally risky and thus get the highest returns. ? 3. Return on the lower part of the frontier are perfectly positively correlated with the discount factor, and hence negatively correlated with the consumption. They provide best insurance against consumption fluctuations. ? 4.All frontier returns are also perfectly correlated with each other. If we pick any single frontier return , then all frontier returns must be expressible as ? for some number a; m R mv R )( fmfmv RRaRR ?+= To be continued 4. Any frontier portfolio carry all information about the discount factor, and hence carry carries all pricing information: there must be exist information a,b,c,d 5. So expected returns can be expressed in a single-beta representation using any mean-variance efficient return (except the risk-free rate) and 6. We can plot the decomposition of a return into a priced or systematic part, and a residual part. Asset inside the frontier is not worse than frontier portfolio. emdRbRam mvmv +=+= ])([)( , fmv mvi fi RRERRE ?+= β fmv RRE ?= )(λ Slope of the mean-standard deviation Frontier and equity premium puzzle ? The ratio of mean excess return to standard deviation is known as Sharp ratio ? The slope of the mean-standard deviation frontier is the largest available Sharp ratio, and thus is naturally interesting. ? For a portfolio in the frontier, the slope of the frontier is ? The slope of the frontier is governed by the volatility of the discount factor. ? If consumption growth is lognormal, consider power utility function ? )( )( i f i R RRE σ ? f mv fmv Rm mE m R RRE )( )( )( ( )( ) σ σ σ == ? )ln(1 )( )( )ln( 1 22 ce R RRE t c mv fmv ?≈?= ? + ? γσ σ σγ Slope of the mean-standard deviation Frontier and equity premium puzzle-to be continued ? The slope of the mean-standard deviation frontier is higher if the economy is riskier—if the consumption is more volatile—or if investors are more risk averse. ? Over last 50 years, the stock returns are averaged 9%, with a standard deviation of about 16%, while the real return on t-bills is 1%. You can conclude that historically annual Sharp ratio was 0.5. Aggregate nodurable and service consumption is 1%. You can deduce that the investors have a risk-aversion coefficient of 50. Random walks and time-varying expected returns ? With the basic pricing formula ? If investors are risk neutral, or if investor’s consumptions have no uncertainty, if security pays no dividend at t+1, for a short time is close to 1, this equation reduces to ? equivalent, prices follow a time-series process of a form ? Prices follow martingales. ? Since consumption and risk aversion don’t change much day to day, random walk view holds pretty well on day to day basis. ? But in the long run, empirical study indicates the long-run excess returns are quite predictable. )])(('[)(' 111 +++ += tttttt dpcuEcup β β )( 1+ = tt pEp 0)( 111 =+= +++ ttttt Ewithpp εε Random walks and time-varying expected returns--- to be continued ? To think about this issue, we write the basic equation as ? The equation indicates that returns can be somewhat predictable--- expected return can change over time. ? First, if conditional variances change over time, we might expect the conditional mean change as well. But empirical research indicates that variables that forecast means do not forecast variances. ? It is not plausible the risk or risk aversion changes at daily basis. ? It is much plausible that risk and risk aversion change at business cycle. ? That is exactly the horizon at which we predict excess returns. ? Models that this connection precise are a very active area of current research. ） 1111 1 11 1 ,()()( )( ),(cov )( ++++ + ++ + ?=?=? tttttttt tt ttt f ttt RmRc mE Rm RRE ρσσγ Present value statement ? The more straightforward way is to write out a long-term objective ? With price , we can buy a stream of cash flows ? The first-order condition give us the pricing formula directly ? ? Because this formula hold for every time t, we can get two-period formula from it ? We can also write a risk adjustment to prices ∑ ∞ = + 1 )( j jt j t cuE β t p { }...3,2,1, = + jd jt ∑∑ ∞ = +++ ∞ = + == 0 , 0 )]('/)('[ j jtjtttjtt j jt j tt dmEdcucuEp β )]([ 111 +++ += ttttt dpmEp ∑∑ ∞ = ++ ∞ = + + += 1 , 1 , ),(cov j jttjtt j f jtt jtt t md R dE p Discount factors in continuous- time ? The choice of discrete versus continuous time is one of modeling convenience. ? Modeling in continuous time often let you have a analytical results that would be unavailable in discrete time. ? In most cases, it would be clear to begin with a discrete model. ? One should be clear enough to discrete and continuous time representation of the same ideas to pick the ideas that is most convenient for a particular application. ? A security’s price is , it pay dividends at a rate of ? The instantaneous total return is ? We model the price of risky asset as diffusions, for example ? t p dtD t dt p D p dp t t t t + dzdt p dp t t (.)(.) σμ += Discount factors in continuous- time---continued ? Think of risk-free asset that has a constant price equal to 1, and pay the risk-free rate as a dividend: ? Or a security that pays no dividend but whose price climbs deterministically at a rate ? The utility function is ? The first-order condition gives us basic pricing equation ? ? Divided by is not a good idea, since the ratio ? is not well behaved for small time interval. f tt rDp == ,1 dtr p dp f t t t = {} ∫ ∞ ? = 0 )()( dtcueEcU t t t δ dsDcueEcup s stst s ttt ∫ ∞ = ++ ? = 0 )(')(' δ ))((' tcu )('/)(' tt cucu ?+ Discount factors in continuous- time---continued ? We define discount factor in continuous time as ? Then we can write pricing equation as ? the analog to the one-period pricing equation is ? Because we will write down price process for and discount process for , we break up using Ito’s lemma, divide (*) by ? ? ? Apply the basic formula to a risk-free asset ? If a risk free asset is not traded, the above formula define a shadow risk free rate or zero-beta rate. )(' t t t cue δ? =Λ ∫ ∞ ++ Λ=Λ 0 dsDEp ststttt )(mxEp = *)]([0 pdEDdt t Λ+Λ= dp Λd )( tt pd Λ (**)][0 p dpd p dpd Edt p D t Λ Λ ++ Λ Λ += Λp )( t t t f t d Edtr Λ Λ ?= Discount factors in continuous- time---continued ? With this interpretation, equation (**) becomes ? With ? Denote the local curvature and third derivative of the utility function as ? For power utility, the former is coefficient , the later is ? Using this relationship, we can write the relationship between risk-free rate and consumption growth as: ? ? ? ? ? ? Λ Λ ?=+ t t t t t f t t t t t t p dpd Edtrdt p D p dp E )( 2 22 )(' )(''' 2 1 )(' )('' t tt t t t tt t t c dc tu tuc c dc cu cuc dt d ++?= Λ Λ δ )(' )(''' )(' )('' tu tu cu cuc t t tt t =?= ηγ γ )1( +γγ )( 1 2 1 )( 1 )( 1 2 2 t t tt t t tt t t t f t c dc E dtc dc E dt d E dt r ηγδ ?+= Λ Λ ?= Discount factors in continuous- time---continued ? We can easily express asset prices in terms of consumption risk rather than consumption risk ? Thus asset returns covary more strongly with consumption get higher mean excess returns. ? ? ? ? ? ? ? ? =?+ t t t t t f t t t t t t p dp c dc Edtrdt p D p dp E γ)(