Discount Rates John H. Cochrane University of Chicago Booth School of Business January 8, 2011
Discount rates 1. Facts: How risk discount rates vary over time and across assets. 2. Theory: Why discount rates vary. I 3. Applications I Macro, Behavioral, Segmented/institutional, Liquidity Portfolio theory, Active/passive management, Accounting, Corporate Finance 4. Apology see long paper for citation, documentation
Forecasting with DP Horizon k b t(b) R 2 σ [E t (R e σ[e )] t (R e )] E (R e ) 1 year 3.8 (2.6) 0.09 5.5 0.76 5 years 20.6 (3.4) 0.28 29.3 0.62 Rt!t+k e = a + b D t P t + ε t+k ; σ [E t (R e )] σ ˆb D t P t 4 x D/P and Annualized Following 7 Year Return 25 4 x DP Return 20 15 10 5 0 1950 1960 1970 1980 1990 2000 2010
Long-Horizon Regression Coe cients and Price Volatility I Identity: (dp t log(d t /P t ); ρ = 0.96) dp t k ρ j 1 r t+j k ρ j j=1 j=1 I Long-run regressions, and coe cient identity 1 d t+j + ρ k dp t+k k ρ j j=1 1 r t+j = a + b r (k) dp t + ε r t+k, etc. ) 1 b (k) r b (k) d + b(k) dp. b (k) r b (k) d b (k) dp Direct regression, k = 15 1.01-0.11-0.11 Implied by VAR, k = 15 1.05 0.27 0.22 VAR, k = 1.35 0.35 0.00 I Why do prices (p/d) move? 100% (135%!) discount rates, 0% (-35%!) dividend growth
A Pervasive Phenomenon, and cycles I A pervasive phenomenon: 1. Stocks. DP! Return, not dividend growth 2. Treasuries. Yield! Return, not rising rates 3. Bonds/CDS. Yield! Return, not default 4. Foreign Exchange. Interest spread! Return, not devaluation 5. Sovereign Debt, Foreign Assets.! Return, not repayment, exports 6. Houses. Price/Rent! Return, not rent growth. I Common element, business cycle association: low prices, high returns in recessions. High prices, low returns in booms I Bubble? "Prices too high" () Discount rate too low
log scale Houses Price and Rent 7.8 20 x rent CSW price OFHEOprice 7.6 Price 7.4 7.2 7 20 x Rent 6.8 1960 1970 1980 1990 2000 2010 Date Houses: b t R 2 Stocks: b t R 2 r t+1 0.12 (2.52) 0.15 0.13 (2.61) 0.10 d t+1 0.03 (2.22) 0.07 0.04 (0.92) 0.02 dp t+1 0.90 (16.2) 0.90 0.94 (23.8) 0.91
A Pervasive Phenomenon, and cycles I A pervasive phenomenon: 1. Stocks. DP! Return, not dividend growth 2. Treasuries. Yield! Return, not rising rates 3. Bonds/CDS. Yield! Return, not default 4. Foreign Exchange. Interest spread! Return, not devaluation 5. Sovereign Debt, Foreign Assets.! Return, not repayment, exports 6. Houses. Price/Rent! Return, not rent growth. I Common element, business cycle association: low prices, high returns in recessions. High prices, low returns in booms I Bubble? "Prices too high" () Discount rate too low
Multivariate Challenges: More variables 1. Many forecasters. Multiple regression? Common forecasters across assets? r stock t+1 = a s + b s dp t + c s ys t + d 0 sz t + ε s t+1? r bond t+1 = a b + c b ys t + b b dp t + d 0 b z t + ε b t+1? 2. Are E t (r i t+1 ) = b i x t correlated across assets? Factor structure of time-varying expected returns? 3. Relate mean to covariance 4. Can t just run big regressions! E t r i t+1 = covt (r i t+1f 0 t+1)λ t 5. Back to prices (price/dividend) long-run forecasts?
Understanding prices. short and long-run forecasts 40 30 2.5 3 E(r lr dp,cay) E(r lr dp) dp 20 3.5 10 0 4 10 dp and cay dp only ac tual r t+1 4.5 20 1950 1960 1970 1980 1990 2000 2010 R t+1 = a + b dp t [+c cay t ] + ε t+1 ; 5 1950 1960 1970 1980 1990 2000 2010 j=1 ρ j 1 r t+j = a + b dp t [+c cay t ] + ε
The cross section 1. Chaos 2. CAPM E (R ei ) = β i E (R em ) 3. Chaos again E (R ei ) = α i + β i E (R em ) (value) 4. Fama and French E (R ei ) = β i E (R em ) + h i E (hml) + s i E (smb) 3. Chaos again E (R ei ) = α i + β i E (R em ) + h i E (hml) + s i E (smb) (Market, value, size), momentum, accruals, equity issues, beta-arbitrage, credit risk, bond & equity market timing, carry trade, put writing, liquidity provision,...
Average return Value e ect and factor 4. Fama and French E (R ei ) = β i E (R em ) + h i E (hml) + s i E (smb) Average returns and betas 0.8 E(r) 0.6 b x E(rmrf) 0.4 β x E(rmrf) 0.2 h x E(hml) 0 0.2 Growth Value Fama - French 10 B/M sorted portfolios..
Value (size, and bond factors) 4. Fama and French E (R ei ) = β i E (R em ) + h i E (hml) + s i E (smb) a. Theories (m) only need to explain the factor E (R ei ) =... + h i E (hml) (Fama French) E (hml) = cov(hml, m) (Theory) b. Value stocks rise and fall together; mean,covariance. (APT). But theories must now explain covariance! c. Value betas explain other E (R e ) sorts, e.g. sales growth. 5. Chaos again..how to repeat FF? E (R ei ) = α i + β i E (R em ) + h i E (hml) + s i E (smb) (Market, value, size), momentum, accruals, equity issues, beta-arbitrage, credit risk, bond & equity market timing, carry trade, put writing, liquidity provision,...
The Multidimensional Challenge I (Market, value, size), momentum, accruals, equity issues, beta-arbitrage, credit risk, bond & equity market timing, carry trade, put writing, liquidity provision,... 1. Which of these are independently important for E (R e )? ( multiple regression ) 2. Does E (R e ) spread correspond to new factors? 3. Do we need all the new factors? Or again, fewer factors than E (R e ) characteristics? 4. Why do prices move? Long run. I How to approach such a highly multidimensional problem?
Asset Pricing on Characteristics/Uni cation 1. Portfolio sorts are really cross-sectional regressions E(R) Portfolio Mean Securities Portfolio 1 2 3 4 5 Log(B/M) E (R ei ) = a + b log(b/m i ) + ε i ; i = 1, 2,...N
Asset Pricing on Characteristics/Uni cation 1. Portfolio sorts are really cross-sectional regressions E (R ei ) = a + b 0 C i + ε i ; i = 1, 2,...N 2. Time series and cross-section are really the same thing R ei t+1 = a + b 0 C it + ε i t+1 3. Result: Expected return is a function of characteristics E (R ei t+1jc it ) C it = [size, b/m, momentum, accruals, d/p, credit spread...] 4. Covariance with factors is also a function of characteristics cov t (Rt+1, ei f t+1 ) = g(c it ) E (R e jc ) = g(c ) λ?
Prices? 1. Why ER/β, not p, PV? 2. Long-run / price in the cross-section? 3. Prices/long run may simplify. 3.1 Campbell-Shiller: 3.2 One-period: ρ j 1 rt+j i = a + b 0 C it + ε i? j=1 ρ j 1 r t+j = ρ j 1 d t+j dp t j=1 j=1 Dt+1 R t+1 = D t+1 = P t r t+1 = d t+1 dp t D t Pt / D t
Theory classi cation 1. Frictionless a. Macroeconomics macro data. i. Consumption ii. Investment iii. Background risks outside income iv. General equilibrium. b. Behavioral Irrational expectations.= discount rate. c. Finance E (R)/β, return-based factors; a ne models. 2. Frictions a. Liquidity. i. Idiosyncratic ii. Systemic iii. Information trading. b. Segmented Di erent investors in di erent markets c. Intermediated Leveraged intermediaries.
Consumption/habits Surplus consumption (C X)/C and stocks P/D SPC (C X)/C 1990 1992 1995 1997 2000 2002 2005 2007 2010 X t k j=0 φ j C t j ; risk aversion t = γ C t C t X t
Investment and Q 4 3.5 Nonres. Fixed I/K and Q I/K P/(20*D) ME/BE 3 2.5 2 1.5 1 1990 1992 1995 1997 2000 2002 2005 2007 2010 1 + α i t k t = market t book t = Q t
Challenges for theories 10 baa,aaa baa aaa and stocks 9 4 baa aaa sp500 p/d 8 7 6 3.5 3 5 4 3 BAA AAA 20 Yr 2.5 2 1.5 2 1 1 5 Yr 0.5 1 Yr 0 2007 2008 2009 2010 2011 2007 2008 2009 2010 Bond yields Bonds and stocks I Pervasive, coordinated risk premium in all markets, especially unintermediated I Mean returns are associated with comovement. I Strong correlation with macroeconomics
Arbitrages Source: Fontana (2010)
Arbitrages Source: Baba and Parker (2008).
Price and volume in the tech bubble. Dollar volume 800 500 700 450 400 600 NASDAQ Tech 350 NASDAQ Tech 500 300 400 250 300 NASDAQ 200 NASDAQ 150 200 100 NYSE 100 NYSE 50 Feb98 Sep98 Mar99 Oct99 Apr00 Nov00 May01 Dec01 0 Feb98 Sep98 Mar99 Oct99 Apr00 Nov00 May01 Dec01 I Price (discount rate) ) Volume? Or some Volume ) Price, like money? I Why so much information trading?
Portfolio theory with many factors I The average investor must hold the market I Portfolio theory based on di erences
bond price Bonds a cautionary tale Price of a bond that matures in year 10 simulation 100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 time, years
log scale percent Percent Stocks (your endowment) in the crisis S&P500 S&P500 volatility 10 0 80 70 montly v ol. v ix What to do?? 10 60 20 50 30 40 40 30 20 50 What to do??? 10 60 2007 2008 2009 2010 2007 2008 2009 2010 share = 1 E (R e ) γ σ 2 (R e ). 0.6 = 1 0.04 2 0.18 2 =) 1 0.04 2 0.70 2 = 0.04???
Prices and payo s: a mean-variance benchmark If utility is quadratic, max fct g E t=0 δ t 1 2 (ct c ) 2 and for any amount of time-varying expected returns, Long run mean Ẽ (x) = 1 1 β j=0 β j E (x t+j )
Alphas, betas, and performance evaluation 50 EquitMktNeut 40 30 20 10 0 10 20 30 40 50 HF rmrf 1994 1996 1998 2000 2002 2004 2006 2008 2010 R ei t = α i + β i rmrf t + h i hml t + s i smb t + u i umd t + vol., carry, beta-arb, issu
Procedures, corporate, accounting, regulation. I Capital budgeting, valuation value of investment = expected payout R f + β [E (R m ) R f ], I Accounting, regulation, capital structure, if prices can change on discount rate news?
Conclusion I Discount rates vary over time and across assets a lot more than you thought I Empirical: how. Theoretical: why. Applications: at all. I We ve only started I How do you ask the right question?
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