Equilibrium Asset Returns

Equilibrium Asset Returns Equilibrium Asset Returns 1/ 38

Introduction We analyze the Intertemporal Capital Asset Pricing Model (ICAPM) of Robert Merton (1973). The standard single-period CAPM holds when investment opportunities are constant, but with changing investment opportunities, a multi-beta ICAPM results. Breeden (1979) shows that this multi-beta ICAPM can be written as a single consumption beta CAPM (CCAPM). We also analyze the general equilibrium production economy model of (Cox, Ingersoll and Ross,1985) that is useful for pricing contingent claims. Equilibrium Asset Returns 2/ 38

ICAPM Model Assumptions Individuals can trade in a risk-free asset paying rate of return of r (t) and in n risky assets whose rates of return are ds i (t) = S i (t) = i (x; t) dt + i (x; t) dz i (1) where i = 1; :::; n, and ( i dz i )( j dz j ) = ij dt. The k state variables follow the process: dx i = a i (x; t) dt + b i (x; t) d i (2) where i = 1; :::; k, and (b i d i )(b j d j ) = b ij dt and ( i dz i )(b j d j ) = ij dt. Equilibrium Asset Returns 3/ 38

Constant Investment Opportunities When r and the i s, i s, and ij s are constants, we showed previously that it is optimal for all individuals to choose the risky assets in the relative proportions k = nx kj ( j r) j=1 nx i=1 j=1 nx ij ( j r) This single risky asset portfolio s mean and variance is nx i i 2 i=1 nx i=1 j=1 (3) nx i j ij : (4) Equilibrium Asset Returns 4/ 38

CAPM Equilibrium Similar to our derivation of the single-period CAPM, we argue that in equilibrium this common risky-asset portfolio must be the market portfolio; that is, = m and 2 = 2 m. Moreover, the continuous-time market portfolio is exactly the same as that implied by the single-period CAPM. Thus, asset returns in this continuous-time environment satisfy the same relationship as the single-period CAPM: where i = im = 2 m. i r = i ( m r) ; i = 1; : : :, n (5) Equilibrium Asset Returns 5/ 38

CAPM Implications Thus, the constant investment opportunity set assumption replicates the standard, single-period CAPM. Yet, rather than asset returns being normally distributed as in the single-period CAPM, the ICAPM has asset returns that are lognormally distributed. While the standard CAPM results continue to hold for this more realistic intertemporal environment, the assumptions of a constant risk-free rate and unchanging asset return means and variances are untenable. Clearly, interest rates vary over time, as do the volatilities of assets such as common stocks. Equilibrium Asset Returns 6/ 38

Stochastic Investment Opportunities For a single state variable, x, the system of n equations that an individual s portfolio weights satisfy are: 0 = A( i r) + nx ij! j W H i ; i = 1; : : : ; n (6) j=1 where A = J W =J WW = U C = [U CC (@C=@W )] and H = J Wx =J WW = (@C=@x) = (@C=@W ). Rewrite (6) in matrix form, and use the superscript p to denote the values for the p th person (individual): A p ( re) =! p W p H p (7) where = ( 1 ; :::; n ) 0, e is an n-dimensional vector of ones,! p = (! p 1 ; :::;!p n) 0 and = ( 1 ; :::; n ) 0. Equilibrium Asset Returns 7/ 38

Aggregate Asset Demands Sum (7) across all persons and divide by P p Ap : re = a h (8) where a P p W p = P p Ap, h P p Hp = P p Ap, and P p!p W p = P p W p is the average investment in each asset. These must be the market weights. The i th row (i th asset excess return) of equation (8) is Pre-multiply (8) by 0 to obtain i r = a im h i (9) m r = a 2 m h mx (10) where mx = 0 is the covariance between the market portfolio and the state variable, x. Equilibrium Asset Returns 8/ 38

Portfolio that Best Hedges the State Variable De ne 1, which are the weights of the portfolio that e 0 1 best hedge changes in the state variable, x. The expected excess return on this portfolio is found by pre-multiplying (8) by 0 : r = a m h x (11) where m and x are the hedge portfolio s covariances with the market portfolio and the state variable, respectively. Solving the two linear equations (10) and (11) for a and h, and substituting them back into equation (9) gives: i r = im x i m 2 m x mx m ( m r)+ i 2 m im mx 2 m x mx m r (12) Equilibrium Asset Returns 9/ 38

ICAPM It can be shown that (12) is equivalent to i r = im 2 i m 2 m 2 2 m ( m r) + i 2 m im m 2 2 m 2 m m i ( m r) + i r (13) where i is the covariance between the return on asset i and that of the hedge portfolio. i = 0 i i = 0. If x is uncorrelated with the market so that m = 0, equation (13) simpli es to r i r = im 2 m ( m r) + i 2 r (14) Equation (13) generalizes to multiple state variables with an additional beta for each state (c.f., APT). Equilibrium Asset Returns 10/ 38

Extension to State-Dependent Utility If an individual s utility is a ected by the state of economy, so that U (C t ; x t ; t), the form of the rst order conditions for consumption (C t ) and the portfolio weights (! i ) remain unchanged and equation (13) continues to hold. The only change is the interpretation of H, the hedging coe cient. Taking the total derivative of envelope condition J W = U C : @C J Wx = U CC @x + U Cx (15) so that H = @C=@x @C=@W U Cx U CC @C @W implying that portfolio holdings minimize the variance of marginal utility. (16) Equilibrium Asset Returns 11/ 38

Breeden s Consumption CAPM Substitute in for A p and H p in equation (7) for the case of k state variables and rearrange to obtain: U p C U p CC C p W ( re) =! p W p + C p x=c p W (17) where C p W = @C p =@W p, C p x = @C p 0, @x 1 ::: @C p @x k and is the n k asset return - state variable covariance matrix whose i,j th element is ij. Pre-multiplying (17) by C p W : U p C U p CC ( re) = W pc p W + Cp x (18) where W p is n 1 vector of covariances between asset returns and investor p s wealth. Equilibrium Asset Returns 12/ 38

Covariance between Asset Returns and Consumption Using Itô s lemma, individual p s optimal consumption, C p (W p ; x; t), follows a process whose stochastic terms for dc p are C p W!p 1 W p 1 dz 1 + ::: +! p nw p n dz n +(b1 d 1 b 2 d 2 :::b k d k ) C p x (19) The covariance of asset returns with changes in individual p s consumption are given by calculating the covariance between each asset (having stochastic term i dz i ) with the terms given in (19). Denoting C p as this n 1 vector of covariances: C p = W pc p W +Cp x (20) Equilibrium Asset Returns 13/ 38

Consumption Covariance The right-hand sides of (20) and (18) are equal, implying: C p = U p C U p CC ( re) (21) De ne C as aggregate consumption per unit time and de ne T as an aggregate rate of risk tolerance where T X p U p C U p CC (22) Then aggregate (21) over all individuals to obtain re = T 1 C (23) where C is the n 1 vector of covariances between asset returns and changes in aggregate consumption. Equilibrium Asset Returns 14/ 38

Excess Expected Returns on Risky Assets Multiply and divide the right-hand side of (23) by current aggregate consumption to obtain: re = (T=C) 1 ln C (24) where ln C is the n 1 vector of covariances between asset returns and changes in log consumption growth. Let a portfolio, m, have weights! m that pre-multiply (24): m r = (T=C) 1 m;ln C (25) where portfolio m s expected return and covariance with consumption growth is m and m;ln C. Portfolio m may or may not be the market portfolio. Equilibrium Asset Returns 15/ 38

Consumption CAPM Use (25) to substitute for (T=C) 1 in (24): re = ( ln C = m;ln C ) ( m r) = ( C = mc ) ( m r) (26) where C and mc are the consumption betas of all asset returns and portfolio m s return. The consumption beta for any asset is de ned as: ic = cov (ds i =S i ; d ln C) =var (d ln C) (27) Equation (26) says that the ratio of expected excess returns on any two assets or portfolios of assets is equal to the ratio of their betas measured relative to aggregate consumption. Hence, the risk of a security s return can be summarized by a single consumption beta. Equilibrium Asset Returns 16/ 38

A Cox, Ingersoll, and Ross Production Economy The ICAPM and CCAPM are not general equilibrium models since they take the form of equilibrium asset price processes as given. However, their asset price process can be justi ed based on the Cox, Ingersoll, and Ross (1985) continuous-time, production economy model. The model assumes that there is a single good that can be either consumed or invested. This capital-consumption good can be invested in any of n di erent risky technologies that produce an instantaneous change in the amount of this good. Equilibrium Asset Returns 17/ 38

Model Assumptions If an amount i is physically invested in technology i, then the proportional change in the good produced is d i (t) i (t) = i (x; t) dt + i (x; t) dz i, i = 1; :::; n (28) where ( i dz i )( j dz j ) = ij dt. The rate of change in the invested good produced has expected value and standard deviation of i and i. Note that each technology displays constant returns to scale and i and i can vary with time and with a k 1 vector of state variables, x(t). Thus, the economy s technologies for transforming consumption into more consumption re ect changing (physical) investment opportunities. Equilibrium Asset Returns 18/ 38

Model Assumptions cont d The i th state variable is assumed to follow the process dx i = a i (x; t) dt + b i (x; t) d i (29) where i = 1; :::; k, and (b i d i )(b j d j ) = b ij dt and ( i dz i )(b j d j ) = ij dt. If each technology is owned by an individual rm, nanced entirely by shareholders equity, then the rate of return on shareholders equity of rm i, ds i (t) =S i (t), equals the proportional change in the value of the rm s physical assets (capital), d i (t) = i (t). Here, ds i (t) =S i (t) = d i (t) = i (t) equals the instantaneous dividend yield where dividends come in the form of a physical capital-consumption good. Equilibrium Asset Returns 19/ 38

Model Assumptions cont d CIR s speci cation allows one to solve for the equilibrium prices of securities other than those represented by the n risky technologies. This is done by imagining there to be other securities that have zero net supplies. Assuming all individuals are identical in preferences and wealth, this amounts to the riskless investment having a zero supply in the economy, so that r is really a shadow riskless rate. Yet, this rate would be consistent, in equilibrium, with the speci cation of the economy s other technologies. Equilibrium Asset Returns 20/ 38

Solving for the Equilibrium Riskless Rate An individual s consumption and portfolio choice problem allocates savings to rms investing in the n technologies. An equilibrium is de ned as a set of interest rate, consumption, and portfolio weight processes fr; C ;! 1 ; :::;! ng such that the individual s rst order conditions hold and markets clear: P n i=1! i = 1 and! i 0. Since the capital-consumption good is physically invested in the technologies, the constraint against short-selling,! i 0, applies. Because, in equilibrium, the representative individuals do not borrow or lend, the situation is exactly as if a riskless asset did not exist. Equilibrium Asset Returns 21/ 38

Optimization without a Riskless Asset Consider the individual s problem as before except that the process for wealth excludes a risk-free asset: Z T max E t U (C s ; s) ds + B(W T ; T ) (30) C s ;f! i;s g;8s;i subject to dw = t nx! i W i dt C t dt + i=1 nx! i W i dz i (31) and P n i=1! i = 1 and! i 0. The rst order condition for consumption is 0 = @U (C ; t) @C i=1 @J (W ; x; t) @W (32) Equilibrium Asset Returns 22/ 38

Optimal Portfolio Weights with Short Sale Constraints Let be the Lagrange multiplier for P n i=1! i = 1. Then the rst order conditions for! i is i @J @W i W + @2 J @W 2 np ij! j W 2 P + k j=1 j=1 @ 2 J @x j @W ijw 0 0 = i! i i = 1; : : : ; n (33) Kuhn-Tucker conditions (33) imply that if i < 0, then! i = 0, so that i th technology is not employed. Assuming (28) and (29) are such that all technologies are employed so i = 0 8i, then the solution is! i = J W J WW W np ij j j=1 kp m=1 j=1 np J Wxm J WW W ij jm + J WW W 2 np ij j=1 (34) Equilibrium Asset Returns 23/ 38

Equilibrium Portfolio Allocation Using matrix notation, (34) becomes! = A W 1 A J W W 2 1 e+ kx j=1 H j W 1 j (35) where A = J W =J WW, H j = J Wxj =J WW, and j = ( 1j ; :::; nj ) 0. The weights are a linear combination of k + 2 portfolios. The rst two portfolios are mean-variance e cient portfolios: 1 is the portfolio on the e cient frontier that is tangent to a line drawn from the origin (a zero interest rate) while 1 e is the global minimum variance portfolio. The last k portfolios, 1 j, j = 1; :::; k, hedge against changes in the technological investment opportunities. Equilibrium Asset Returns 24/ 38

Equilibrium Consumption and Portfolio Weights The proportions of these k + 2 portfolios chosen depend on the individual s utility. An exact solution is found in the usual manner of substituting (35) and C = G (J W ) into the Bellman equation. For speci c functional forms, a value for the indirect utility function, J (W ; x; t) can be derived. This, along with the restriction P n i=1! i = 1, determines the speci c optimal consumption and portfolio weights. Since in the CIR economy the riskless asset is in zero net supply, portfolio weights in (35) must be those chosen by the representative individual even if o ered the opportunity to borrow or lend at rate r. Equilibrium Asset Returns 25/ 38

Portfolio Allocation with a Riskless Asset Recall that portfolio weights for the case of including a risk-free asset are! = A kx W 1 H j ( re) + W 1 j ; i = 1; : : : ; n (36) j=1 (35) and (36) are identical when r = = (J W W ). Hence, r = WJ W (37) =! 0 W A!0! + kx j=1 H j A!0 j (37) is the same as (10) extended to k state variables. Hence, the ICAPM and CCAPM hold for the CIR economy. Equilibrium Asset Returns 26/ 38

Valuing Contingent Claims in Zero Net Supply The CIR model also can be used to nd the equilibrium shadow prices of other contingent claims. Suppose the payo of a zero-net-supply contingent claim depends on wealth, time, and the state variables, so that P (W ; t; fx i g). Itô s lemma implies where dp = updt + P W W P n i=1! i i dz i + P k i=1 P x i b i d i (38) up = P W W! 0 C + P k i=1 P x i a i + P t + P WW W 2! 0! 2 + P k i=1 P Wx i W! 0 i + 1 2 P k i=1 P k j=1 P x i x j b ij (39) Equilibrium Asset Returns 27/ 38

Expected Return under ICAPM The ICAPM relation (9) extended to k states is u = r + W A Cov (dp=p; dw =W ) P k i=1 H i A Cov (dp=p; dx i ) (40) Noting that the representative agent s wealth is the market portfolio: up = rp + 1 A Cov (dp; dw ) P k H i i=1 A Cov (dp; dx i ) = rp + 1 P W W 2! 0! + P k i=1 A P x i W! 0 i P k i=1 H i A P W W! 0 i + P k j=1 P x j b ij (41) Equilibrium Asset Returns 28/ 38

Equilibrium Partial Di erential Equation Equating (39) and (41) and recalling the value of the equilibrium risk-free rate in (37), we obtain a partial di erential equation for the contingent claim s value: 0 = P WW W 2! 0! + 1 P k P k i=1 j=1 2 2 P x i x j b ij + P k i=1 P Wx i W! 0 i + P W (rw C) + P t P k i=1 P W x i a i A!0 i + P k H j b ij j=1 rp (42) A Equilibrium Asset Returns 29/ 38

An Example with Log Utility Let U(C s ; s) = e s ln (C s ) and B (W T ; T ) = e T ln (W T ). Then we previously showed that J (W ; x; t) = d (t) e t ln (W t ) + F (x; t) where d (t) = 1 1 (1 ) e (T t), so that Ct = W t =d (t). Since J Wxi = 0, H i = 0, and A = W, the portfolio weights in (35) are! = 1 ( re) (43) where r = = (J W W ). Equilibrium Asset Returns 30/ 38

Market Portfolio Weights Market clearing requires e 0! = 1, and solving for r(t): r = e0 1 1 e 0 1 e (44) Substituting (44) into (43),the equilibrium portfolio weights are: e! = 1 0 1 1 e 0 1 e (45) e Equilibrium Asset Returns 31/ 38

Square Root Process State Variable Assume that a single state variable, x (t) ; a ects all production processes: d i = i = b i x dt + b i p xdzi, i = 1; :::; n (46) where b i and b i are assumed to be constants and the state variable follows the square root process where dz i d = i dt. dx = (a 0 + a 1 x) dt + b 0 p xd (47) If a 0 > 0 and a 1 < 0, x is a nonnegative, mean-reverting random variable. Equilibrium Asset Returns 32/ 38

Equilibrium Interest Rate Process Write the technologies n 1 vector of expected rates of return as = bx and their n n matrix of rate of return covariances as = b x. Then from (44), the equilibrium interest rate is r = e0 b 1 b 1 e 0 b x = x (48) 1 e where e 0 b 1 b 1 =e 0 b 1 e is a constant. Thus, the risk-free rate follows the square root process: dr = dx = (r r) dt + p rd (49) where a 1 > 0, r a 0 =a 1 > 0, and b 0 p. When 2r 2, if r (t) > 0 it remains positive at all future dates, as would a nominal interest rate. Equilibrium Asset Returns 33/ 38

Valuing Default-Free Discount Bonds Consider the price of a default-free bond that matures at T t. P W, P WW, and P Wx in (42) are zero, and since r = x, the bond s price can be written as P (r; t; T ). The PDE (42) becomes 2 r 2 P rr + [ (r r) r] P r rp + P t = 0 (50) where b! 0. b b! equals the right-hand side of equation (45) but with replaced by b and replaced by, b while b is an n 1 vector whose i th element is b i i. r =! 0 is the covariance of interest rate changes with (market) wealth, or the interest rate s beta. Equilibrium Asset Returns 34/ 38

Solution for Bond Prices Solving PDE (50) subject to P (r; T ; T ) = 1 leads to where = T P (r; t; T ) = A () e B()r (51) q t, ( + ) 2 + 2 2, " 2e (++ ) 2 A () ( + + ) (e 1) + 2 B () # 2r = 2 2 e 1 ( + + ) (e 1) + 2 (52) (53) Note that the bond price is derived from an equilibrium model of preferences and technologies rather than the absence of arbitrage (c.f., Vasicek). Equilibrium Asset Returns 35/ 38

Equilibrium Bond Price Process Note that Itô s lemma implies the bond price follows dp = P r dr + 1 2 P rr 2 rdt + P t dt (54) 1 = 2 P rr 2 r + P r [ (r r)] + P t dt + P r p rd From (50) 1 2 P rr 2 r + P r [ (r dp=p = r r)] + P t = rp + rp r, so that 1 + P r dt + P r P P p rd (55) = r (1 B ()) dt B () p rd where from (51) we substituted P r =P = B (). Equilibrium Asset Returns 36/ 38

Equilibrium Market Price of Interest Rate Risk Hence, p (r; ) p (r; ) r = rb () p rb () = p r (56) Thus we see that the market price of interest rate risk is proportional to the square root of the interest rate. When < 0, which occurs when the interest rate is negatively correlated with the return on the market portfolio (and bond prices are positively correlated with the market portfolio), bonds will carry a positive risk premium. Equilibrium Asset Returns 37/ 38

Summary The Merton ICAPM shows that when investment opportunities are constant, the expected returns on assets satisfy the single-period CAPM In general, an asset s risk premia include the asset s covariances with asset portfolios that hedge against changes in investment opportunities. The multi-beta ICAPM can be simpli ed to a single consumption beta CCAPM. The Cox, Ingersoll, and Ross model shows the ICAPM results are consistent with a general equilibrium production economy. The model also is used to derive the equilibrium interest rate and the shadow prices of securities in zero net supply. Equilibrium Asset Returns 38/ 38