Advanced Modern Macroeconomics

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1 Advanced Modern Macroeconomics Asset Prices and Finance Max Gillman Cardi Business School 0 December 200 Gillman (Cardi Business School) Chapter 7 0 December 200 / 38

2 Chapter 7: Asset Prices and Finance Chapter Summary Dynamic exogenous growth model extended consumer invests in capital directly plus government bonds. Price of government bonds equals marginal product of capital. Then rm directly invests in capital instead of consumer: who buys goods producer shares, receives dividend pro t. Bond return equals goods share return, given no uncertainty. In nite horizon includes a limiting condition on stock price called a "transversality condition". Stock price value a simple relation to rm s capital stock. Assuming constant dividends stock price is discounted value of in nite stream of dividends, reducing to just dividend divided by interest rate. The equity premium: di erence between risky stock return and risk free return on bonds. Derived and explained with exogenous, endogenous growth. Gillman (Cardi Business School) Chapter 7 0 December / 38

3 Building on the Last Chapters Develops investment into capital found previously. Part 4 : consumer directly invests in capital, rents to rm. Chapter 6: consumer saves in bank which lends to rm. Now nancial markets introduced: bonds and stocks: consumer invests directly through nancial markets. Firm direct investment in capital, also found in last chapter. But all other chapters: rm rents capital from consumer. Unlike previously, now uncertainty in capital investment. Makes shares in rm risky, higher than government bond return. Now a risk premium exists, related to Chapter 5 uncertainty. Exogenous growth as in Part 4, but fails to explain empirical ndings of equity premium. So use endogenous growth of Part 5 to improve this. Gillman (Cardi Business School) Chapter 7 0 December / 38

4 Learning Objective Asset pricing fundamentals: government bonds, shares in rms. Bond, share price, equal capital marginal product without uncertainty. Why share price returns exceed bond price return under uncertainty. Equity premium: focus on di erence is key, big focus in macroeconomics nance. literature. Other focus: share price re ects discounted dividend stream. Gillman (Cardi Business School) Chapter 7 0 December / 38

5 Who Made It Happen Asset prices in general equilibrium representative agent model: attributed to Lucas 978 "Asset Prices in an Exchange Economy." Called "consumption based capital asset pricing model." Built upon vast work in nance theory, without full general macroeconomic equilibrium. 990 Nobel Prize: nance pioneers Markowitz, Miller, Sharpe. Fama 970: stock prices re ect information available to market. Mehra, Prescott 985 "The equity premium: A puzzle": see if basic dynamic macroeconomic model explains equity premium. Stimulated giant literature to answer result that model fails: reviewed by Kocherlakota 996; still ongoing additions to literature. Cochrane 200 expands macroeconomics link to nance theory. Recent: using human capital, Zhang 2006 explains equity premium. Gillman (Cardi Business School) Chapter 7 0 December / 38

6 Modeling the Return on Bonds Assume same Cobb-Douglas goods production, log utility. Government "nominal" bonds, of quantity b t, Interest rate return on bonds: R t. Nominal bonds" return R t is market price: without money, in ation, nominal return equals real return, R t will be same as real return r t δ k to capital. Money introduced in Part 7, with in ation possibly a ecting R t. At time t, consumer invests in bonds for next period:b t+, receives interest income plus investment in bonds back. Interest income is R t b t, plus get quantity b t : total b t ( + R t ). Minus outlay b t+, total net bond investment is b t+ b t ( + R t ). Gillman (Cardi Business School) Chapter 7 0 December / 38

7 Baseline Dynamic Exogenous Growth Model Consumer invests in physical capital directly: V (k t ) = y t = A G l γ t k γ t = c t + i t, i t = k t+ ( δ k ) k t, = x t + l t, c t = w t l t + r t k t k t+ + k t ( δ k ). V (k t ) = Max : ln c t + α ln x t + βv (k t+ ), c t,x t,l t,k t+ Max : ln [w t l t + r t k t k t+ + k t ( δ k )] + a ln ( l t ) + βv (k t+ ). l t,k t+ Now add investment in government bonds, plus consumer receipt of lump sum transfer G t : government budget constraint is G t = b t+ b t ( + R t ). Gillman (Cardi Business School) Chapter 7 0 December / 38

8 Consumer Investment in Government Bonds c t = w t l t + r t k t k t+ + k t ( δ k ) + G t b t+ + b t ( + R t ). V (k t, b t ) = Max l t,k t+,b t+ ln [w t l t + G t k t+ + k t ( + r t δ k ) b t+ + b t ( + R t )] +α ln ( l t ) + βv (k t+, b t+ ) ; 0 = w t + α ( ), ( ) + β V (k t+, b t+ ), c t x t c t k t+ V (k t, b t ) k t = u (c t, x t ) c t ( + r t δ k ), β + ρ, =) c t = + r t δ k, w t = c t + ρ r t = ( γ) A G l γ t k γ t, w t = γa G l γ t k γ t. Gillman (Cardi Business School) Chapter 7 0 December / 38 α x t c t ;

9 Plus Equilibrium Conditions for Bonds b t+ : 0 = u (c t, x t ) c t ( ) + β V (k t+, b t+ ) b t+, b t : V (k t, b t ) b t = u (c t, x t ) c t ( + R t ), =) 0 = u (c t, x t ) c t ( ) + β V (k t, b t ) b t, V (k t, b t ) b t = u (c t, x t ) β c t, c t = + R t, β βc t + ρ, c t c t = + r t δ k c t + ρ =) R t = r t δ k. u(c t,x t ) c t β u(c t,x t ) c t = + R t ; c t = + R t + ρ ; Gillman (Cardi Business School) Chapter 7 0 December / 38

10 Example 7. Alternative Bond Pricing Alternative bond pricing: look at price of asset rather than yield on asset. Price is inverse of gross yield. Instead of bond investment b t+ b t ( + R t ), write as ˆq t+ b t+ b t, where ˆq t+ is price of bonds b t+. Maximization of utility with respect would imply q t = =. + r t δ k + R t One dollar of a bond bought at discount less than one: +R t. Upon maturation at end of period, pay o is b t. So yield so-called "discount bond" would be R t. Gillman (Cardi Business School) Chapter 7 0 December / 38

11 Equity Capital Asset Market Under certainty assumptions, rm s ownership shares yield same as bond, so consumer indi erent; government bond and equity shares have same return. When risk in rm s divident payout on equity shares, then di erence in return on risky equity, risk-free bonds. Government bonds like from US Treasury viewed as risk-free, but only an abstraction; never completely risk-free. The Equity Premium: return di erence between risky equity, risk-free bond. Requires extending model with pricing of risky equity shares. Denote equity return by Rt+ S, government bond return R t+. Use expectations with E t being the expected value at time t. De ne equity premium by i E t hr t+ S R t+. Must determine E t R S t+ in general equilibrium. Gillman (Cardi Business School) Chapter 7 0 December 200 / 38

12 A Model of Asset Prices under Certainty Equity return model with perfect foresight, no uncertainty. Denote share of goods producer by s t, with price pt S ; time t net investment: pt S s t+ + pt S s t; dividends dt S s t : c t = w t l t + G t b t+ + b t ( + R t ) p S t s t+ + V (b t, s t ) = Max l t,b t+,s t+ h ln w t l t + G t b t+ + b t + b t R t pt S s t+ + +α ln ( l t ) + βv (b t+, s t+ ) ; s t+ : 0 = u (c t, x t ) c t s t : p S t p S t + d S t + β V (b t+, s t+ ) s t+, V (b t, s t ) s t = u (c t, x t ) c t (p s t + d s t ) ; =) + R t = [u (c t, x t )] / c t β [ u (c t, x t ) / c t ] s t, i pt S + dt S s t = ps t + d S t p S t + R s t. Gillman (Cardi Business School) Chapter 7 0 December / 38

13 Capital Gain from Price Change, Plus Dividend Return to stocks ps t +d s t p s t : stock price rate of increase pt s pt s plus dividend yield, known commonly as " nancial capital gain", Equivalent formulation: d s t p s t. + R S t = ps t + d s t = ln ln ( + x) = x, p s t = + ps t p s t p s t + ps t p s t p s t + ln + d s t + d s t p s t p s t ;, R S t ps t p S t p S t + d S t p S t. Net equity return equals sum of "capital price gain" plus dividend. Gillman (Cardi Business School) Chapter 7 0 December / 38

14 Goods Producer Dynamic Problem Dividends s t d S t are residual rm pro t Revenues minus costs are output Alt γ k t, minus labor cost w t l t, investment k t+ k t ( δ k ), plus proceeds of equity share sales.pt S s t+ pt s s t. s t d S t = A G l γ t k γ t w t l t k t+ + k t ( δ k ) + p S t s t+ p s t s t. Single rrepresentative consumer: shares equal, s t = s t+ =, =) p S t s t+ p S t s t = p S t p S t = 0; d S t = A G l γ t k γ t w t l t k t+ + k t ( δ k ). Dynamic recursive maximization problem discount rate: consumer s intertemporal rate, β u(c t+,x t+) c t+ u(ct,xt ) ct = +R t+. Gillman (Cardi Business School) Chapter 7 0 December / 38 γ

15 Recursive Framework Goods producer problem: 0 p S (k t ) = Max l t,k t+ = Max l t,k t+ = Max l t,k t+ S t + β u(c t+,x t+ ) c t+ u(c t,x t ) c t A p S (k t+ ) A dt S + ps (k t+ ) + R t+ Al γ t k γ t w t l t k t+ + k t ( δ k ) + ps (k t+ ) + R t+ R t = ( γ) A G lt k t r t ( γ) A G lt k t γ δ k, γ, R t = r t γ lt δ k ; w t = γa G. k t Gillman (Cardi Business School) Chapter 7 0 December / 38

16 Solving Goods Producer Equilibrium Equity Value p S (k t ) = A G l γ t k γ t w t l t k t+ + k t ( δ k ) + ps (k t+ ) + R t+, y t = A G l γ t k γ t = w t l t + r t k t, p S (k t ) = w t l t + r t k t w t l t k t+ + k t ( δ k ) + ps (k t+ ) + R t+, p S (k t ) = k t ( + r t δ k ) k t+ + ps (k t+ ) + R t+. p S (k t+ ) = k t+ ( + r t+ δ k ) k t+2 + ps (k t+2 ) + R t+2 ; p S (k t ) = k t ( + r t δ k ) k t+ + k t+ ( + r t+ δ k ) k t+2 + ps (k t+2) +R t+2. + R t+ Gillman (Cardi Business School) Chapter 7 0 December / 38

17 Equity Price as Capital Value p S (k t ) = k t ( + r t δ k ) k t+ + k t+ ( + r t+ δ k ) k t+2 + ps (k t+2) +R t+2, + R t+ = k t ( + r t δ k ) p S (k t ) = k t ( + r t δ k ) k t+2 p + S (k t+2 ) + R t+ ( + R t+ ) ( + R t+2 ) ; + lim j! p S (k j ) ( + R t+ ) ( + R t+2 ) ( + R t+j ) ; 0 = lim j! p S (k j ) ( + R t+ ) ( + R t+2 ) ( + R t+j ) ; p S (k t ) = k t ( + r t δ k ) "Ex-dividend" price: ˆp S (k t ) = k t. Gillman (Cardi Business School) Chapter 7 0 December / 38

18 Limiting Discounted Price: Transversality Condition Discounted value of price, t!, is zero called "transversality condition"; assumes that p S k j lim = 0. j! ( + R t+ ) ( + R t+2 ) + R t+j Implies no price "bubble" where it can rise forever above fundamental value, which is discounted stream of dividends. Transversality condition assumed implicitly not to hold when bubbles exist in steady state equilibrium. Gillman (Cardi Business School) Chapter 7 0 December / 38

19 Example 7.2 Stock Price, Capital Value, Zero Growth p S (k t ) = k t ( + r t δ k ) = k t ( + g) ( + ρ), g = 0, =) p S (k t ) = k t ( + ρ). Chapter 6 drop in value of capital stock k t due to drop in bank productivity A F compares to drop in value of stock market. Stock market value represented by p S (k t ). If k t falls, then same as stock market value falling. In this simple model with zero exogenous growth. Gillman (Cardi Business School) Chapter 7 0 December / 38

20 Asset Pricing under Uncertainty p S t V (b t, s t ) = Max l t,b t+,s t+ E t fln [w t l t b t+ + b t ( + R t ) pt s s t+ + (pt s + dt s ) s t ] +α ln ( l t ) + βv (b t+, s t+ )g ; w t = αc t u (c t, x t ) V (bt+, s t+ ), = βe t x t c t b t+ V (bt+, s t+ ), u (c t, x t ) c t = βe t Envelope : s t+ V (b t, s t ) = u (c t, x t ) ( + R t ), b t c t V (b t, s t ) = u (c t, x t ) (pt s + dt s ). s t c t, Gillman (Cardi Business School) Chapter 7 0 December / 38

21 Basic Asset Pricing Condition Combining all equilibrium conditions, 2 pt S = E t 4 β u(c t+,x t+ ) c t+ u(c t,x t ) c t p S t+ + d S t+3 5. Price: expected discounted next period price plus dividend. Discount rate reduces to + R t+, since 2 = E t 4 β u(c 3 t+,x t+ ) c t R t+ u(c t,x t ) c t Nominal interest rate for next period in general unknown, but as "risk-free" bond return of R t+ is known, so also is the expected discounted intertemporal margin. Gives basic asset pricing relation: pt S = E t + R t+ p S t+ + d S t+. Gillman (Cardi Business School) Chapter 7 0 December / 38

22 Example 7.3 Asset Pricing with Covariance Let two terms of asset pricing condition be called X t +R t+, Y t p S t+ + ds t+. Expectation is product of terms plus their covariance: p S t = E t [X t Y t ] = E t [X t ] E t [Y t ] + Cov t [X t Y t ]. Non-zero covariance term introduces risk premium. Example 7.4. Expectations Hypothesis Suppose Cov t [X t Y t ] = 0, =) p S t = E t p S t = E t + R t+ i h pt+ S + ds t+. + R t+ h i E t pt+ S + ds t+, Price: expected future price plus dividend without risk premium. Gillman (Cardi Business School) Chapter 7 0 December / 38

23 Prices as Dividend Stream under Zero Covariance pt S = E t p S t+ + dt+ S, pt+ S = E t+ pt+2 S + d S t+2 ; + R t+ + R t+2 (p S E t E t+2 +dt+2) S t+ pt s +R t+2 + dt+ S = + R t+ = E " t d S t+ pt+2 S + E t E + d # t+2 S t+ + R t+ ( + R t+ ) ( + R t+2 ) = E " t d S t+ E t+2 pt+3 S + E t E + d S # t+3 t+ + R t+ ( + R t+ ) ( + R t+2 ) ( + R t+3 ) " # dt+2 S +E t E t ( + R t+ ) ( + R t+2 ) Gillman (Cardi Business School) Chapter 7 0 December / 38

24 Asset Price Reduction as In nite Dividend Stream Given transversality condition: p lim S (k j) j! (+R t+ )(+R t+2 )(+R = 0. t+j) and E t E t+ E t+2 = E t E t+ E t+2 E t+3 =... = E t, price is just discounted dividend stream: h i pt s = E t dt+ S " + E t E + t+ R t+ " +E t E t+ E t+2 =) p S t = d S t+2 ( + R t+ ) ( + R t+2 ) dt+3 S ( + R t+ ) ( + R t+2 ) ( + R t+3 ) p S k j +... lim ; j! ( + R t+ ) ( + R t+2 ) + R t+j 2 3 E t j= 6 4 j i =0 dt+j S 7 5. ( + R t+i ) Gillman (Cardi Business School) Chapter 7 0 December / 38 # #

25 Example 7.5 Constant Average Dividend Stream Assume constant dividend, constant interest rate over time.! =) p S = d S + R + + R + ( + R) 2 + ( + R) ,! p S d = S = d S + R R. +R Return: dividend rate d S divided by share price p S, R = ds p S. Uncertainty: dividend stream stochastic over time. Consider. "risk-adjusted" average real interest rate R S average expected risky dividend stream ˆd S : p S = ˆd S. R S p Price-earnings ratio: S ˆd =. S R S Gillman (Cardi Business School) Chapter 7 0 December / 38

26 Balanced Exogenous Growth Path, No Uncertainty d S t = A Gt l γ t k γ t w t l t i t, dt+ S dt S pt S = pt S = = y t+ w t+ l t+ i t+ y t w t l t i t = y t ( + g) w t ( + g) l t i t ( + g) y t w t l t i t = + g. dt S + R + d t S ( + g) d S t + R ( + R) 2 + d t S ( + g) ( + g) ( + R) , A = d S t + R + R ( + R) ( + g) (+g ) (+R ), p S t = d S t R g. Price of equity higher when growth rate higher: US in 990s. Gillman (Cardi Business School) Chapter 7 0 December / 38

27 Endogenous Growth d S t = A G (l t h t ) γ k γ t w t l t h t i t, p S t (k) = k t h t ( + r t δ k ). Example 3.2 : A H increases by 5%; g = 0.02 to g = ; r t = to k t h t falls from k t h t = to k t h t = 0.694; no uncertainty, h t =, p S t (k) = k ( + r t δ k ) =.0 ( ) =.0737, falls: to p S t (k) = k ( + r t δ k ) = ( ) = But h t may rise as k t falls. Wealth may rise is h t rises while k t falls: W t = w t h t T t ρ(+g ) + k t; human capital a much bigger fraction of wealth than k t. Wealth division between capitals: helps resolve equity premium puzzle. Gillman (Cardi Business School) Chapter 7 0 December / 38

28 Deriving the Equity Premium "Consumption-based capital asset pricing model", or CCAPM : p S t = E t p S t = E t = 2 4 β u(c t+,x t+ ) c t+ u(c t,x t ) 2 c t 4 β u(c t+,x t+ ) c t+ u(c t,x t ) c t 3 (pt+ s + dt+) s 5 ; 3 h i 5 E t pt+ S + dt+ S 20 +Cov t 4@ β u(c t+,x t+ ) c t+ A pt+ S + dt+ 3 S 5 ; u(c t,x t ) c t E t p S t+ + d S t+ ( + R t+ ) pt 20 S +Cov t 4@ β u(c t+,x t+ ) c t+ A ps t+ + d S t+ p S t! 3 5. u(c t,x t ) c t Gillman (Cardi Business School) Chapter 7 0 December / 38

29 Equity Premium as Covariance Term E t p S t+ + dt+ S pt S ( + R t+ ) 20 = ( + R t+ ) Cov t 4@ β u(c t+,x t+ ) c t+ A, u(c t,x t ) c t p S t+ + d S t+ p S t! 3 5 ; + E t hr S t+ E t hr S t+ i i E t p S t+ + d S t+ p S t R t+ = ( + R t+ ) Cov t!, β u(c t+,x t+ ) c t+, + R S 5 u(c t,x t ) t+. c t Gillman (Cardi Business School) Chapter 7 0 December / 38

30 Simpli cation of Equity Premium with Log Utility i E t hr t+ S i E t hr t+ S i E t hr t+ S ct R t+ = ( + R t+ ) βcov t, + Rt+ S, c t+ β = + ρ, + Rt+ ct R t+ = Cov t, + Rt+ S, + ρ c t+ R t+ ' ρ; ct R t+ = Cov t, + Rt+ S. c t+ Gillman (Cardi Business School) Chapter 7 0 December / 38

31 Example 7.7 Empirical Failing of Exogenous Growth Empirical covariance-variance relationship: ct Cov t, + Rt+ S ct Var t Var t + Rt+ S. c t+ c t+ Computation for US: Average equity premium: 0.06, or 6%, variance of consumption: 0.036, variance ofexcess equity return: 0.67, Var ct t c t+ Var t + Rt+ S = (0.036) (0.67) = Result: computed equity premium 0 times lower than 6%. Model fails to explain equity premium data. Gillman (Cardi Business School) Chapter 7 0 December / 38

32 Equity Premium and Human Capital c t = w t h t + α g + δ y Pt = H w t h t A H W t y Pt R W t = w th t g + δ H A H R W t W t = y Pt = w t h t W t+ = c t = + Rt W + α + k t ρ ( + g) + k t ρ ( + g) ; c t = g +δ H, + α A H + k t ρ ( + g) Rt W, g + δ H + k t ρ ( + g), A H W t c t, y Pt = + α R W t W t. y Pt. Gillman (Cardi Business School) Chapter 7 0 December / 38

33 Human Capital Model Equity Premium and Data W t c t = W t Rt W W t, + α W t+ W t c t = Wt + Rt W W t +α R W t = W t + R W t. c t W t +α +α R W t R W t = + α + αrw t + α R W t R m t. Zhang 2006: R m t essential part that enters equity premium; estimates Rt m with more general model, wealth decomposition into human, non-human capital: human capital much less volatile; and stock market equity only 6% share of consumer wealth. Zhang: covariance of R m t with the equity return is large: implies plausible equity premium puzzle explanation. Gillman (Cardi Business School) Chapter 7 0 December / 38

34 Why Adding Human Capital Can Explain Equity Premium Human capital generally not fully capitalized. Including human capital resolves puzzle because return on stock market equity only a part of return R W t, Lower volatilaty of human capital also basis of Jagannathan, Kocherlakota 996: why people switch from stocks to bonds when older. Asset pricing equation would hold without human capital modi cation if all of consumer s wealth perfectly traded in asset markets, including human capital. But human capital imperfectly traded in markets, through labor contracts, insurance prices, causes equity premium equation to appear to fail. Adjusting for imperfect trading in human capital, Zhang nds consistency of model with equity premium data. Gillman (Cardi Business School) Chapter 7 0 December / 38

35 Application: Historical Price Earnings Ratios Stock and Bond data: US Standard and Poors 500 rm index, in blue, Nominal long term US government bond rate, in red, supplied by Professor Robert Shiller s website. Can consider how changes over historical period in price-earnings ratio can be explained in terms of war, business cycles and tax rates. Next graph: Blue US Price-earnings on right vertical axis; Red US bond market interest rate on left vertical axis; year on horizontal axis, from 880 to Gillman (Cardi Business School) Chapter 7 0 December / 38

36 Price Earnings Ratio Long Term Interest Rates Year Gillman (Cardi Business School) Chapter 7 0 December / 38

37 Possible Role of War Unusual acceleration, high peak of price-earnings ratio (P-E) in 920s, in 990s: share major preceding event, end of major international war with new long term peaceful prospects. Turning down of these two peaks both coincided with new large-scale international wars: WWII, Terrorism war. End of WWI brought expansion of US capital markets abroad prospect of higher earnings, although with great uncertainty. 929 stock price crash could have re ected lack of expected earnings and prospect of lower earnings from oncoming new WWII. After WWII, new war started: Cold War of US, Western Europe versus Soviet Union, China; including 950s Korean War, Vietnam. Eastern Europe, continental Asia, excluded from capital markets; Western Europe, certain "Paci c Rim" Asia countries included. Full post WWII expansion delayed until post-989 Cold War end. Huge upturn in US asset markets in 990s, Crash of equity markets after 2000, in 2008: after onset of war in Central, Middle-Eastern Asia. Gillman (Cardi Business School) Chapter 7 0 December / 38

38 Possible Business Cycle, Tax Role Cycles in price-earnings might be correlated with business cycle. P-E higher early in economic expansions, lower early in contractions. as higher future dividends in expansion built into equity price, before earnings materialize; opposite in recession. 980 US downturn and 970s correlated with very high interest rates. But are "nominal" interest rates re ecting high in ation. In ation in next Part 7 of text. "In ation tax" can reduce growth, lower expected dividends. Drop in P-E from 960s to 98 could be from rising in ation tax, from 960s to 980, rather than business cycles. Looks like growth trend change more than business cycle factors; trends in growth rate, capital return, a ected by major tax levels. Gillman (Cardi Business School) Chapter 7 0 December / 38