Discount Rates

John H. Cochrane

University of Chicago Booth School of Business

April 11, 2011

Discount rates

1. Facts: How risk discount rates vary over time and acrossassets.

2. Theory: Why discount rates vary.I “Macro,”“Behavioral,”“Segmented/institutional,”“Liquidity”

3. ApplicationsI Portfolio theory, Active/passive management, Accounting,Corporate Finance

4. Apology — see long paper for citation, documentation

Forecasting with DPHorizon k b t(b) R2 σ [Et (Re )]

σ[Et (R e )]E (R e )

1 year 3.8 (2.6) 0.09 5.5 0.765 years 20.6 (3.4) 0.28 29.3 0.62

Ret→t+k = a+ bDtPt+ εt+k ; σ [Et (Re )] ≡ σ

(b× Dt

Pt

)

1950 1960 1970 1980 1990 2000 2010

0

5

10

15

20

25

4 x D/P and Annualized Following 7­Year Return

4 x DPReturn

Long-Horizon Regression Coeffi cients and Price VolatilityI Identity: (dpt ≡ log(Dt/Pt ); ρ = 0.96)

dpt ≈k

∑j=1

ρj−1rt+j −k

∑j=1

ρj−1∆dt+j + ρkdpt+k

I Long-run regressions, and coeffi cient identity

k

∑j=1

ρj−1rt+j = a+ b(k )r × dpt + εrt+k , etc.

⇒ 1 ≈ b(k )r − b(k )∆d + b(k )ρkdp .

b(k )r b(k )∆d b(k )ρkdp

Direct regression , k = 15 1.01 -0.11 -0.11Implied 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. 0% “Rational Bubble.”

A Pervasive Phenomenon, and cycles

I A pervasive phenomenon:

1. Stocks. DP → Return, not dividend growth2. Treasuries. Yield → Return, not rising rates3. Bonds/CDS. Yield → Return, not default4. Foreign Exchange. Interest spread → Return, not devaluation5. Sovereign Debt, Foreign Assets. → Return, not repayment,exports

6. Houses. Price/Rent → Return, not rent growth.

Houses —Price and Rent

1960 1970 1980 1990 2000 2010

6.8

7

7.2

7.4

7.6

7.8

20 x Rent

Price

Date

log 

scal

e

20 x rentCSW priceOFHEOprice

Houses: b t R2 Stocks: b t R2

rt+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) 0.02dpt+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 growth2. Treasuries. Yield → Return, not rising rates3. Bonds/CDS. Yield → Return, not default4. Foreign Exchange. Interest spread → Return, not devaluation5. 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, lowreturns in booms

I “Bubble?”"Prices too high" ⇐⇒ Discount rate “too low”

Multivariate Challenges: More variables

1. Many forecasters. Multiple regression? Common forecastersacross assets?

r stockt+1 = as + bs × dpt + cs × yst + d ′szt + εst+1?

rbondt+1 = ab + bb × dpt + cb × yst + d ′bzt + εbt+1?

2. Are Et (r it+1) = bi × xt correlated across assets? Factorstructure of time-varying expected returns?

3. Relate mean to covariance

Et(r it+1

)= covt (r it+1f

′t+1)λt

4. Can’t just run big regressions!

5. Back to prices (price/dividend) — long-run forecasts?

Understanding prices. Short and long-run forecasts

Coeffi s t-stats Otherdpt cayt dpt cayt R2 σ [Et (yt+1)]

rt+1 0.12 0.071 (2.14) (3.19) 0.26 8.99∆dt+1 0.024 0.025 (0.46) (1.69) 0.05 2.80dpt+1 0.94 -0.047 (20.4) (-3.05) 0.91cayt+1 0.15 0.65 (0.63) (5.95) 0.43

r lrt 1.29 0.033 0.51∆d lrt 0.29 0.033 0.12

r lrt =∞

∑j=1

ρj−1rt+j ; ∆d lrt =∞

∑j=1

ρj−1∆dt+j

Understanding prices. Short and long-run forecasts

1950 1960 1970 1980 1990 2000 2010­20

­10

0

10

20

30

40

dp and caydp onlyac tual  r

t+ 1

rt+1 =a+ b× dpt [+c × cayt ] + εt+1;

1950 1960 1970 1980 1990 2000 2010­5

­4.5

­4

­3.5

­3

­2.5E(r l r|dp,cay)E(r l r|dp)dp

∑∞j=1 ρj−1rt+j =

a+ b× dpt [+c × cayt ] + ε

dpt =∞

∑j=1

ρj−1rt+j −∞

∑j=1

ρj−1∆dt+j

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

R es pons e to∆d s hoc k

returndiv  grow ths hoc k  da te

0 5 10 15 20­0.05

0

0.05

0.1

0.15R es pons e to  dp s hoc k

Σρj­1 rt+j

=1.29

Σρj­1 ∆dt+j=0.29

rt = ­0 .96

returndiv  grow th

0 5 10 15 20­0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

R es pons e to  1σ c ay  s hoc k

Σρj­1 rt+j=0.033

Σρj­1 ∆dt+j=0.033

returndiv  grow th

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

R es pons e to∆d s hoc k

pr ic ediv idends hoc k  da te

0 5 10 15 20

­1

­0.8

­0.6

­0.4

­0.2

0

0.2

R es pons e to  dp s hoc k

pr ic ediv idend

0 5 10 15 20

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16R es pons e to  1σ c ay  s hoc k

pr ic ediv idend

The “cross section”

1. Chaos

2. CAPM E (Rei ) = βiE (Rem)

3. Chaos again E (Rei ) = αi + βiE (Rem) (value)

4. Fama and French

E (Rei ) = βiE (Rem) + hiE (hml) + siE (smb)

3. Chaos again

E (Rei ) = αi + βiE (Rem) + hiE (hml) + siE (smb)

(Market, value, size), momentum, accruals, equity issues,beta-arbitrage, credit risk, bond & equity market timing, carrytrade, put writing, “liquidity provision,”...

Value effect and factor4. Fama and French

E (Rei ) = βiE (Rem) + hiE (hml) + siE (smb)

Grow th Value­0.2

0

0.2

0.4

0.6

0.8E(r)

β x  E(rmrf)

b x  E(rmrf)

h x  E(hml)

Aver

age 

retu

rnAv erage returns  and betas

Fama - French 10 B/M sorted portfolios. .

Value (size, and bond factors)4. Fama and French

E (Rei ) = βiE (Rem) + hiE (hml) + siE (smb)

a. Theories (m) only need to explain the factor

E (Rei ) = ...+ hiE (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 (Re ) sorts, e.g. sales growth.

5. Chaos again..(Market, value, size), momentum, accruals,issues, beta-arbitrage, credit risk, bond & equity markettiming, carry trade, put writing, “liquidity provision,..

E (Rei ) = αi + βiE (Rem) + hiE (hml) + siE (smb)

6. How to repeat FF?

The Multidimensional Challenge

I (Market, value, size), momentum, accruals, equity issues,beta-arbitrage, credit risk, bond & equity market timing, carrytrade, put writing, “liquidity provision,”...

1. Which of these are independently important for E (Re )?(“multiple regression”)

2. Does E (Re ) spread correspond to new factors in cov?

3. Do we need all the new factors? Or again, fewer factors thanE (Re ) characteristics?

4. Why do prices move? —Long run.

I How to approach such a highly multidimensional problem?

Asset Pricing on Characteristics/Unification

1. Portfolio sorts are really cross-sectional regressions

Log(B/M)

E(R)

1 2 3 4 5Portfolio

PortfolioMean

Securities

E (Rei ) = a+ b log(b/mi ) + εi ; i = 1, 2, ...N

Asset Pricing on Characteristics/Unification

1. Portfolio sorts are really cross-sectional regressions

E (Rei ) = a+ b′Ci + εi ; i = 1, 2, ...N

2. Time series and cross-section are really the same thing —apanel.

Reit+1 = a+ b′Cit + εit+1

3. Result: Expected return is a function of characteristics

E (Reit+1|Cit )

Cit = [size, b/m, momentum, accruals, d/p, credit spread....]

4. Covariance with factors is also a function of characteristics.

covt (Reit+1, ft+1) = g(Cit )

E (Re |C ) = g(C )× λ?

The value/size premium as a panel regression

sizet bmt ∆sizet ∆bmt1. Cross section -0.030 0.272. Pooled -0.022 0.553. Time dummies -0.031 0.294. Portfolio dummies -0.087 1.485. Pooled -0.030 0.46 -0.38 1.11

FF 25 size and BM portfolios, monthly data.

Reit+1 = a+ (at ) + (ai ) + b× sizei ,t + c × bmi ,t+ (d × ∆12sizei ,t + e × ∆12bmi ,t ) .

Prices?

1. Why ER/β, not p,PV ?2. Prices/long run may simplify.

2.1 Campbell-Shiller:

∞

∑j=1

ρj−1rt+j =∞

∑j=1

ρj−1∆dt+j − dpt

2.2 One-period:

Rt+1 =Dt+1Pt

=

(Dt+1Dt

)/(PtDt

)rt+1 = ∆dt+1 − dpt

3. Long-run / price in the “cross-section”?

∞

∑j=1

ρj−1r it+j = a+ b′Cit + εi ?

Panel regressions and prices —simple example

Portfolio dummies Time dummies Pooledb b

1−ρφ b b1−ρφ dpt−1 ∆dpt−1

r 0.107 0.903 0.044 0.325 0.090 0.074∆d -0.005 -0.041 -0.083 -0.611 -0.004 0.073∆d∗ -0.011 -0.097 -0.092 -0.675 -0.012 0.076dp 0.92 0.90 0.94 0.002

FF 10 size portfolios, monthly data

r it+1 = a+ [ai ] + [at ] + b× dpit +[c × ∆dpit

]+ εit+1

dpit+1 = adp + φdpit + εdpit+1

Theory classification

1. Theories based on fundamental investors, with few frictions.

1.1 Macroeconomics — tie to macro data.

1.1.1 Consumption, Aggregate risks.1.1.2 Risk sharing/background risks (Hedging outside income)1.1.3 Investment and production.1.1.4 General equilibrium, including macroeconomics

1.2 Behavioral Irrational expectations. Tie to price data. Otherdata?

1.3 Finance. ER − β. Price data.

2. Theories based on frictions.

2.1 Segmented markets —different investors in different markets;limited risk bearing.

2.2 Intermediated markets..2.3 Liquidity. i) Idiosyncratic ii) Systemic iii) Information trading.

Risk-neutral probability theorem.

p = ∑s

πsmsxs =1R f ∑

sπ∗s xs

π∗s ≡ πsmsR f

ms = βu′(cs )u′(c)

Investor

Investor

Investor

Security class

Investor Investor

Investor

Investor

Intermediary

“Debt”“Equity”

?

?

Other assets

Segmented markets

Intermediated markets

Security class

Securities

Consumption/habits

1990 1992 1995 1997 2000 2002 2005 2007 2010

Surplus consumption (C­X)/C and stocks

SPC (C­X)/C

P/D

Xt ≈ k ∑∞j=0 φjCt−j ; risk aversiont = γ Ct

Ct−Xt

Investment and Q

1990 1992 1995 1997 2000 2002 2005 2007 20101

1.5

2

2.5

3

3.5

4Nonres. Fixed I/K and Q

I/KP/(20*D)ME/BE

1+ α itkt =markettbookt

= Qt

Challenges for theories

2007 2008 2009 2010 20110

1

2

3

4

5

6

7

8

9

10

BAA

AAA

20 Yr

5 Yr

1 Yr

baa,aaa

Bond yields

2007 2008 2009 2010

0.5

1

1.5

2

2.5

3

3.5

4

baa­aaa and stocks

baa­aaasp500p/d

Bonds and stocks

I Coordinated risk premium in all markets, incl. unintermediatedI Mean returns are associated with comovement.I Strong correlation with macroeconomics.I Macro: the recession is the risk premium.

“Arbitrages”

Feb 07 Jun 07 Sep 07 Dec 07 Apr 08 Jul 08 Oct 080

1

2

3

4

5

6

Bond

CDS

Perc

ent

Source: Fontana (2010)

“Arbitrages”

Feb 07 Jun 07 Sep 07 Dec 07 Apr 08 Jul 08 Oct 08 Jan 09 May 090

1

2

3

4

5

6

7

swaplibor

3 Month rates. Source: Baba and Packer (2009).

Price and volume in the tech “bubble.”

Feb98 Sep98 Mar99 Oct99 Apr00 Nov00 May01 Dec01

50

100

150

200

250

300

350

400

450

500

NYSE

NASDAQ

NASDAQ Tech

Feb98 Sep98 Mar99 Oct99 Apr00 Nov00 May01 Dec010

100

200

300

400

500

600

700

800

NYSE

NASDAQ

NASDAQ Tech

Dollar volume

I Price (discount rate) ⇒ Volume? Or some Volume ⇒ Price,like money?

I Why so much information trading?

Portfolio theory with many factorsI Sexy profit ideas? The average investor must hold the marketI Portfolio theory based on differences

Bonds —a cautionary tale

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Price of a bond that matures in year 10 ­­ s imulation

time, years

bond

 pric

e

Stocks (your endowment) in the crisis

2007 2008 2009 2010­60

­50

­40

­30

­20

­10

0

10

What to do???

log 

scal

e ­­ 

perc

ent

S&P500

2007 2008 2009 2010

10

20

30

40

50

60

70

80

What to do??

Perc

ent

S&P500 volati l ity

montly  vol.v ix

share =1γ

E (Re )σ2(Re )

. 0.6 =120.040.182

=⇒ 120.04

0.702= 0.04???

Payoff streamsI For long-run investors the coupons of an indexed perpetuityare the risk free asset, not the short term rate!

I Risky asset? If utility is quadratic,max{ct} E ∑∞

t=0 δt(− 12)(ct − c∗)2 and for any amount of

time-varying expected returns,

“Long run mean” E (x) = 11−β ∑∞

j=0 βjE (xt+j )

Alphas, betas, and performance evaluation

1994 1996 1998 2000 2002 2004 2006 2008 2010­50

­40

­30

­20

­10

0

10

20

30

40

50EquitMk tNeut

HFrmrf

Reit = αi + βi rmrft +hihmlt + si smbt +uiumdt + vol., carry, beta-arb, issues, ...+εit???

Procedures, corporate, accounting, regulation.

I Capital budgeting, valuation

value of investment =expected payout

R f + β [E (Rm)− R f ] ,

I Accounting, regulation, capital structure, if prices can changeon discount rate news?

I Macroeconoimcs. Recessions are rises in the risk premium,not “the” interest rate. Risk bearing, not allocation ofconsumption over time. .

Conclusion

I Discount rates vary over time and across assets a lot morethan you thought

I Empirical: how. Theoretical: why. Applications: at all.I We’ve only startedI How do you ask the right question?