Asset Price Dynamicswith Slow-Moving Capital

American Finance Association Address

Darrell DuffieStanford University

Atlanta, January, 2010

Acknowledgements: Adam Ashcraft, Nicolae Garleanu, Gaston Giroux, Jeremy

Graveline, Gustavo Manso, Semyon Malamud, Lasse Pedersen, Bruno Strulovici,

Tong-Sheng Sun, Kevin Wu

Time

Price

Efficient market price

Stationary illiquid market price

With limited capital mobility and supply shock

0 10 20 30 40 50 60 70 80 90 100−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Days Since Effective Deletion Date

Exc

ess

Ret

urn

Mean Excess Return for S&P 500 Deletions

Figure 1: Cumulative returns for dropped S&P500 stocks.

−1

−3

−5

−7

−9

−11

Pric

e

Time0 20 40 60 80 100

Figure 2: Modeled price path with slow investors

Figure 3: An over-the-counter market.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

8.85

8.9

8.95

9

9.05

Pric

e, lo

w λ

Low λHigh λ 9.52

9.57

9.62

9.67

9.72

Pric

e, h

igh

λ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.04

0.08

0.12

Calendar Time

Inst

anta

neou

s R

etur

n

Low λHigh λ

Figure 4: Source: Duffie, Garleanu, and Pedersen (2007).

Figure 5: A centralized market.

Figure 6: A hybrid market structure.

Capital per unit riskMarket 1

Capital per unit riskMarket 2

Price

of risk

Price

of risk

p1

q1

p2

q2

Figure 7: Capital migrates from markets with low risk premia to markets with high

risk premia. Duffie and Strulovici (2008).

1990 1992 199819961994 2000 2002

0

5

10

15

20

25

30

35 140

120

100

80

60

40

20

0

CAMARES price index of

CatXL covers

Insurance industry's annual natural

catastrophe loss burden in USD billion

Figure 8: Catastrophe risk: premiums and global volume of claims. Source: Swiss

Re.

Figure 9: Reinsurance risk premia by line. Source: Swiss Re.

43

-800

-600

-400

-200

0

200

400

1/7/

05

4/7/

05

7/7/

05

10/7

/05

1/7/

06

4/7/

06

7/7/

06

10/7

/06

1/7/

07

4/7/

07

7/7/

07

10/7

/07

1/7/

08

4/7/

08

7/7/

08

10/7

/08

1/7/

09

4/7/

09

7/7/

09

CDS Basis (in basis points) for Investment Grade (bold) and Speculative Grade Bonds. Source: Mitchell and Pulvino (2009)

Excess profits from CIP arbitrage (short dollar spot positions)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.308

-Mar

-06

08-M

ay-0

6

08-J

ul-0

6

08-S

ep-0

6

08-N

ov-0

6

08-J

an-0

7

08-M

ar-0

7

08-M

ay-0

7

08-J

ul-0

7

08-S

ep-0

7

08-N

ov-0

7

08-J

an-0

8

08-M

ar-0

8

08-M

ay-0

8

08-J

ul-0

8

08-S

ep-0

8

08-N

ov-0

8

08-J

an-0

9

08-M

ar-0

9

(pps

)

Long EURUSD spot

Short USDJPY spot

Long GBPUSD spot

Short USDCHF spot

source: Mancini-Griffoli and Ranaldo (2009)

Empirical Evidence from Supply-Shock Price Reactions

∙ Catastrophe risk insurance claims: Froot and O’Connell (1999).

∙ Fire sales from mutual fund redemptions: Coval and Stafford (2007).

∙ Impact of Ford-GM downgrade forced sale of bonds: Feldhutter (2009).

∙ Equity market-order imbalances: Andrade, Chang, and Seasholes (2005),

Hendershott and Seasholes (2006).

∙ Index recompositions: Chen, Noronha, and Singhal (2004), Greenwood

(2005), Mitchell, Pulvino, Stafford (2004).

∙ Stock and bond issuances: Newman and Rierson (2004), Kulak (2008),

Chaiserote (2008).

∙ Various financial-crisis broken arbitrages: Mitchell, Pedersen, and Pulvino

(2007), Collin-Dufresne (2009), Mancini-Griffoli and Ranaldo (2009), Mitchell

and Pulvino (2009).

A Sample of Related Theory

∙ Price behavior with intermediated asynchronous demands: Grossman and

Miller (1987), Brunnermeier and Pedersen (2009).

∙ Price reactions to supply shocks with search delays: Duffie, Garleanu,

Pedersen (2007), Duffie and Strulovici (2008).

∙ Stationary equilibrium with delayed trade: Garleanu (2009), Rosu (2009).

∙ Endogenous investor inattention: Duffie and Sun (1990); Abel, Eberly,

Panageus (2009).

4.0%

3.0%

2.0%

1.0%

0.0%

-1.0%

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18

Cum

ula

tive

retu

rn Index rebalancing

No index rebalancing

Figure 10: Cumulative stock return of acquirer around merger closing. Index rebal-

ancing (red) applies if the acquirer is in the S&P 500 and the target is not. Source:

Mitchell, Pulvino, and Stafford (2004).

5%

0%

-5%

-10%

-15%

-20%

-25%

-30%

Cum

ulat

ive

retu

rnMonths from supply shock

0 2 4 6 8 10 12-4-6-8-10-12-14

Forced buys (sells) CAAR

Figure 11: “Fire sales.” Source: Coval and Stafford (2007).

Equilibrium Model of Supply Shocks with Slow Investors

∙ A fraction q of investors are “inattentive” for k periods after each

trade.

∙ Every period, 1/k of the inattentive investors trade.

∙ The remaining investors trade every period.

∙ The asset supply Zt and dividend process Xt are jointly

Gaussian and autoregressive.

∙ All investors have additive exponential utility.

t k t 1t 2t

t kD

1t kD

2t kD

( )1 2 3 2 1..., , , , ,t t t t k t k t kD DH D D D− − − + − + − +=

( )1 2 3 2 1..., , , , ,t t t t k t k t kD DH D D D− − − + − + − +=

( )1 1 2 3 2, , , ... , ,t t t t tt k kDDH D D D+ − − − + − +=

Stationary Model Solution

∙ Let Ht = (Dt−1, Dt−2, . . . , Dk−1) be the vector of quantities

held off the market by slow investors.

∙ The state vector is Yt = (Zt, Xt, Ht).

∙ Price: St = c ⋅ Yt.

∙ Frequent investor demand: Kt = b(c) ⋅ Yt.

∙ Inattentive-investor demand: Dt = a(c) ⋅ Yt.

∙ Dynamics: Yt+1 = A(c)Yt +B�t. So,

E(Yt+k ∣Yt) = A(c)k Yt.

∙ Market clearing: Dt +Kt = Zt − 1 ⋅Ht ≡ g ⋅ Yt.

∙ Solve the market-clearing equation a(c) + b(c) = g for c.

0 10 20 30 40 50 60 70 80 90 100−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Days Since Effective Deletion Date

Exc

ess

Ret

urn

Mean Excess Return for S&P 500 Deletions

Figure 12: Cumulative returns for dropped S&P500 stocks.

−1

−3

−5

−7

−9

−11

Pric

e

Time0 20 40 60 80 100

Figure 13: A random supply shock occurs on date 1.

20 40 60 80 100

k = 32, q = 0.8k = 24, q = 0.6k = 16, q = 0.4

Pric

e

Time

Figure 14: A random supply shock occurs on date 1.

−20 0 20 40 60−2

0

2

4

6

8

10

12

14Other−Bond Yield−Spread Reaction

Days Relative to Issuance Date

Bas

is P

oint

s

Figure 15: Capital immobility in the Telecom debt market Source: Newman-

Rierson (2003).

−20 −15 −10 −5 F O +5 +10 +15 +20

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

trading days

aver

age

pric

e

avg. rel. mkt. priceavg. rel. offer price

Figure 16: Average price dynamics around secondary equity issuances. Source:

Jan Peter Kulak (2008).

20 40 60 80 100

k = 32, q = 0.8k = 32, q = 0.2

Pric

e

Time

Figure 17: On date 1, a block sale is announced to occur on date 32.

0 1 2 3 4 5 6 7 8 9

47

48

49

50

51

52

53

54

Figure 18: Pre-supply-shock limit orders to buy.

0 1 2 3 4 5 6 7 8 9

47

48

49

50

51

52

53

54

Figure 19: Only the solid-red limit orders remain after the shock.

0 1 2 3 4 5 6 7 8 9

47

48

49

50

51

52

53

54

Figure 20: New (blue) limit orders arrive.

0 1 2 3 4 5 6 7 8 9

47

48

49

50

51

52

53

54

Figure 21: Another market order arrives.

0 2 4 6 8 10 1247

48

49

50

51

52

53

54

55

0 2 4 6 8 10 1247

48

49

50

51

52

53

54

55

Figure 22: Some supply shocks are ”fundamental.”

Perspective

∙ Asset prices can move away from “fundamental” values if capital

is not perfectly mobile.

∙ More precisely, the fundamentals include state variables

determining the immediate and future availability of capital.

∙ Models based on imperfect search are especially natural for

dealer-intermediated and over-the-counter markets.

∙ Capturing trading delays is crucial for any almost any market.

∙ The time signatures of price responses to supply shocks helpidentify capital immobility.