Liquidity and Congestion
Gara Mínguez Afonso London School of Economics
Workshop in Economic Theory, UCLA 8th May 2008
Gara Mínguez Afonso
Liquidity and Congestion
Overview
¦ Liquidity as a coordination phenomenon.
¦ Thick market externalities:
Investors provide market liquidity and market liquidity attracts new investors
¦ However . . .
Workshop in Economic Theory, UCLA
1
Gara Mínguez Afonso
Liquidity and Congestion
¦ . . . as investors concentrate on one side of the market, the market becomes “congested” and trade more difficult.
¦ Congestion reduces the returns to participating in one-sided markets and discourages potential investors from entering.
Workshop in Economic Theory, UCLA
2
Gara Mínguez Afonso
Liquidity and Congestion
An alternative view of market liquidity
¦ Arrival of potential investors causes two opposite effects: 1. Coordination: reduces trading costs (attracts new investors) 2. Congestion: reduces returns to participating in this market (discourages new investors)
¦ Market liquidity depends on which of the two effects dominates. When congestion is the dominating effect. . .
Workshop in Economic Theory, UCLA
3
Gara Mínguez Afonso
Liquidity and Congestion
Results:
1. Reducing market frictions can decrease market attractiveness.
2. Diminishing market frictions can deteriorate market liquidity and reduce welfare.
3. Market illiquidity (measured by the price discount) can increase while trading volume rises.
Workshop in Economic Theory, UCLA
4
Gara Mínguez Afonso
Liquidity and Congestion
My contribution ¦ Complement the literature on self-fulfilling liquidity by incorporating a second effect: the congestion effect.
¦ Market liquidity results from the tradeoff between market complementarities and the congestion effect.
¦ When congestion is the dominating effect I find some interesting results: Diminishing market frictions can deteriorate liquidity and reduce welfare
Workshop in Economic Theory, UCLA
5
Gara Mínguez Afonso
Liquidity and Congestion
The Model ¦ Time is continuous (t ∈ [0, ∞)). ¦ One asset: pays dividend flow d and is in total supply S. ¦ Agents (investors): – Infinitely lived risk-neutral investors. – Hold {0,1} units of the asset. – Time preferences (constant discount rate r > 0). – 3 types of investors in market + Outside investors. Workshop in Economic Theory, UCLA
6
Gara Mínguez Afonso
Liquidity and Congestion
Types of investors Liquidity shock Nonsearcher
Sellerto-be
h0
hs
Buy 1 unit of the asset
Sell the asset + Exit the market
hb
Liquidity shock OUTSIDE INVESTORS
Buyerto-be Enter the market
Figure 1: Measures of investors.
Workshop in Economic Theory, UCLA
7
Gara Mínguez Afonso
Liquidity and Congestion
The Model ¦ Market entry: – Investors are heterogeneous in investment opportunities κ. – The flow of investors entering the economy is defined by f . (f (κ) continuous and strictly positive with support: [κ, κ])
– Only a fraction ν(κ) invests in this market ⇒ Total flow of Rκ investors entering the market: g = κ ν(κ)f (κ)dκ.
¦ Liquidity shocks arrive with Poisson rate γ. (Reduces investors’ valuation from d to d − x, where x > 0)
Workshop in Economic Theory, UCLA
8
Gara Mínguez Afonso
Liquidity and Congestion
The Model ¦ Market operates through search: Vayanos and Wang (2007) 1. Investors meet at rates ληs, ληb: – Overall flow of meetings ληbηs. – Meetings always result in trade. 2. Investors bargain over price p of the asset: z . – Prob(buyer-to-be makes offer)= 1+z
– Take-it-or-leave-it offer. Workshop in Economic Theory, UCLA
9
Gara Mínguez Afonso
Liquidity and Congestion
Equilibrium
Step 1. Measure of Investors
Step 2. Expected Utilities and Price
Step 3. Endogenous Entry Rule
Workshop in Economic Theory, UCLA
10
Gara Mínguez Afonso
Liquidity and Congestion
Equilibrium Step 1. Measure of Investors:
¦ Market clearing: η0 + ηs = S
⇒
ηs = S − η0
(1)
¦ Inflow-outflow conditions (steady state): ληbηs = γη0
⇒
g = γηb + ληbηs
⇒
Workshop in Economic Theory, UCLA
γ η0 λÃS − η0 ! γ 1 η0 g =γ 1+ λ S − η0 ηb =
(2) (3)
11
Gara Mínguez Afonso
Liquidity and Congestion
Measure of Investors
g ×h0 Nonsearcher
Sellerto-be
h0
hs
lhbh s
lhbh s hb
g × hb OUTSIDE INVESTORS
Buyerto-be g
Figure 2: Flows of investors.
Workshop in Economic Theory, UCLA
12
Gara Mínguez Afonso
Liquidity and Congestion
Equilibrium Proposition 1 There is a unique solution to the system (1) (3) given by: η0 =
1 A 2γ
(4)
1 ηs = S − A 2γ A γ ηb = λ 2γS − A
where A = (g + γS +
γ2 λ)−
Workshop in Economic Theory, UCLA
r
(g + γS +
γ2 2 λ)
(5) (6)
− 4γgS.
13
Gara Mínguez Afonso
Liquidity and Congestion
Comparative statics (measures of investors)
¦ The measures of every type of investor change in – the flow of new investors entering the market as follows ∂ηb ∂η0 , >0 ∂g ∂g
and
∂ηs <0 ∂g
and
∂η0 >0 ∂λ
– their search abilities ∂ηb ∂ηs , <0 ∂λ ∂λ
Workshop in Economic Theory, UCLA
14
Gara Mínguez Afonso
Liquidity and Congestion
Step 2. Expected Utilities and Price
g Nonsearcher
d-x
d
v0
-p
vs
lh s
vb Buyerto-be
Sellerto-be
lhb
g
+p
0 OUTSIDE INVESTORS
Figure 3: Flow of utilities and price.
Workshop in Economic Theory, UCLA
15
Gara Mínguez Afonso
Liquidity and Congestion
Expected Utilities and Price
¦ Flow value conditions: rvb = −γvb + ληs(v0 − vb − p)
(7)
rv0 = d + γ(vs − v0)
(8)
rvs = d − x + ληb(p + 0 − vs)
(9)
¦ Bilateral bargaining: p=
z 1 vs + (v0 − vb) 1+z 1+z
Workshop in Economic Theory, UCLA
(10)
16
Gara Mínguez Afonso
Liquidity and Congestion
Proposition 2 The system of equations (7)-(10) has a unique solution given by:
vb = v0 = vs = p =
x ληsz k r + γ (r + γ + ληs)z + γ ¶ µ d x x γ − k +k r r (r + γ + ληs)z + γ r + γ µ ¶ x x d − k +k r r (r + γ + ληs)z + γ x d −k r r
(11) (12) (13) (14)
(r + γ + ληs)z + γ where k = . (r + γ + ληs)z + (r + γ + ληb)
Workshop in Economic Theory, UCLA
17
Gara Mínguez Afonso
Liquidity and Congestion
Comparative statics (price)
¦ The price of the asset is – increasing in the dividend flow and the measure of buyersto-be: ∂p ∂p , >0 ∂d ∂ηb – decreasing in the the buyer’s-to-be bargaining power and the measure of sellers-to-be: ∂p ∂p , <0 ∂z ∂ηs Workshop in Economic Theory, UCLA
18
Gara Mínguez Afonso
Liquidity and Congestion
Step 3. Endogenous Entry Rule ¦ Investors choose market comparing vb to valt (valt(κ) = κ). ¦ The fraction ν(κ) of investors who enters our market is: 0
ν(κ) = [0, 1] 1
κ > κ0 κ = κ0 κ < κ0
if if if
¦ The total flow of investors entering this market is: g(κ0) =
Z κ κ
ν(κ)f (κ)dκ ≡ Proportion of investors(κ < κ0)
¦ In equilibrium: Ã vb g ∗ =
Z κ κ
Workshop in Economic Theory, UCLA
!
ν ∗(κ)f (κ)dκ = valt(κ∗)
(15)
19
Gara Mínguez Afonso
Liquidity and Congestion
Definition 1 A market equilibrium consists of a fraction ν(κ) of investors entering the market, measures (ηs, ηb, η0) of investors and expected utilities and prices (vb, v0, vs, p) such that: • (ηs∗, ηb∗, η0∗ ) solve the market-clearing condition and inflowoutflow equations given by the system (1) - (3), • (vb∗, v0∗ , vs∗, p∗) solve the flow-value equations for the expected utilities and the pricing condition given by the system (7) (10), • ν ∗(κ) solves the entering condition given by equation (15). Workshop in Economic Theory, UCLA
20
Gara Mínguez Afonso
Liquidity and Congestion
Theorem 1 There is a unique market equilibrium. Given valt , vb
valt
∂vb ∂κ0
≤ 0,
∂valt ∂κ0
> 0 and vb > valt |κ0=0 ,
then κ∗ is unique. Given unique κ∗ , g ∗ is unique. Given unique g ∗ , then
vb
k
k1'
k 2' k *
k
- ηs , ηb , η0 unique (Prop. 1) k'
- vs , vb , v0 , p unique (Prop. 2) Equilibrium is unique.
Figure 4: κ∗ defines the outside investment opportunity which makes investors indifferent between entering or not our market.
Workshop in Economic Theory, UCLA
21
Gara Mínguez Afonso
Liquidity and Congestion
Market Frictions, Flow of Investors and Liquidity
¦ What are the consequences on market liquidity of a reduction in market frictions?
¦ We need to consider two effects:
Workshop in Economic Theory, UCLA
22
Gara Mínguez Afonso
Liquidity and Congestion
¦ But the equilibrium flow of investors results from the tradeoff between: – Market complementarities: decreasing trading costs, attracts new investors (increasing the flow).
– Congestion effects: one-sided markets discourage new investors from entering this market (diminishing the flow).
Workshop in Economic Theory, UCLA
23
Gara Mínguez Afonso
Liquidity and Congestion
Market Liquidity ¦ Liquidity increases in the flow of investors entering the market. – Asset price: p = dr − k xr (r+γ+ληs )z+γ – Illiquidity discount: k xr , where k = (r+γ+λη s )z+(r+γ+ληb )
∂k ∂k < 0, >0 ∂ηb ∂ηs ∂k <0 ∂g
Workshop in Economic Theory, UCLA
and
⇒
∂ηs ∂ηb > 0, <0 ∂g ∂g ∂p >0 ∂g
24
Gara Mínguez Afonso
Liquidity and Congestion
An Example of a Technological Innovation ¦ We consider a search-based model of asset trading to examine the consequences of the introduction of a new electronic trading system. ¦ Assumption f ∼ fbeta(κ, a = 1, b = 1)[κ = 0, κ = 5] ≡ U[0, 5] ¦ Parameter values: r = 0.01 S=2 Workshop in Economic Theory, UCLA
d=2 x = 0.4d
γ = 0.2 z=1 25
Gara Mínguez Afonso
Liquidity and Congestion
Example: Baseline Setting (a)
(b)
4
λ=1 λ=2 λ=6 λ=20
valt
3.5 3
0.39
0.38
2
g*
v
b
2.5
1.5
0.37
0.36
1 0.35
0.5 0 0
1
2
3
κ’
4
5
6
0.34 1
2
3
4
5
λ
6
7
8
9
10
Figure 5: Improving efficiency (higher value of λ) attracts more investors to our market (higher g ∗ ).
Workshop in Economic Theory, UCLA
26
Gara Mínguez Afonso
Liquidity and Congestion
Example: Baseline Setting (a)
(b) 160
η*
b
1.5
p*, v*0, v*s
η*b, η*0, η*s
2
η*
0
η*
1
s
0.5 0 1
2
3
4
5
λ
6
7
8
9
s
145 2
3
4
5
2
3
4
5
λ
6
7
8
9
10
6
7
8
9
10
2
1
1.9
0.9
v*b
η*b / η*s
0
v*
150
140 1
10
p* v*
155
0.8 0.7
1.8 1.7
0.6 1
2
3
4
5
λ
6
7
8
9
10
1.6 1
λ
Figure 6: Equilibrium measures of investors and ratio of buyers-to-be to sellers-to-be (a), expected utilities and price (b) as a function of efficiency λ.
Workshop in Economic Theory, UCLA
27
Gara Mínguez Afonso
Liquidity and Congestion
Example: Market Boom (Crash in an Outside Market)
(a)
(b)
1
1
0.8
0.8
f
1.2
f
1.2
0.6
0.6
0.4
0.4
0.2
0.2
0 0
1
2
3
κ
4
5
6
0 0
1
2
3
κ
4
5
6
Figure 7: Beta Distribution: (a = 1, b = 1) (a) and (a = 2, b = 15) (b).
Workshop in Economic Theory, UCLA
28
Gara Mínguez Afonso
Liquidity and Congestion
Example: Crash in an Outside Market (a)
(b)
4
0.6
λ=1 λ=2 λ=6 λ=20
3.5 3
0.58 0.56 0.54
2
g*
vb
2.5
valt
0.52
1.5
0.5
1
0.48
0.5
0.46
0 0
0.5
1
κ
1.5
2
0.44
2
4
6
8
10
λ
12
14
16
18
20
Figure 8: Improving efficiency (higher value of λ) discourages investors from entering our market (lower g ∗).
Workshop in Economic Theory, UCLA
29
Gara Mínguez Afonso
Liquidity and Congestion
Why reducing frictions makes it less attractive to investors? (a)
(b) η*
b
1.5
η*
1
η*
p*, v*0, v*s
η*b, η*0, η*s
2
0 s
0.5 0 1
2
3
4
5
λ
6
7
8
9
180
v*
0 s
175
2
3
4
5
2
3
4
5
λ
6
7
8
9
10
6
7
8
9
10
0.6
4
v*b
η*b / η*s
p* v*
170 1
10
6
2 1
185
0.55 0.5
2
3
4
5
λ
6
7
8
9
10
1
λ
Figure 9: Equilibrium measures of investors and ratio of buyers-to-be to sellers-to-be (a), expected utilities and price (b) as a function of efficiency λ.
Workshop in Economic Theory, UCLA
30
Gara Mínguez Afonso
Liquidity and Congestion
Example: Crash in an Outside Market
¦ Result 1 Decreasing market frictions can reduce the flow of investors moving into our market.
¦ The reason for this counterintuitive result is that lower trading frictions in a one-sided market magnify the effect of congestion, discouraging investors from entering this market.
Workshop in Economic Theory, UCLA
31
Gara Mínguez Afonso
Liquidity and Congestion
Market Frictions and Flow of Investors
¦ Interested in general relationship between the equilibrium flow of investors g ∗ entering the market and market efficiency λ.
¦ Indifference condition: vb = valt ≡ κ ¶ ¶ µ µ x γ −1 = F (S − ηs ) γ 1 + beta 2 r + γ λzηs + [(r + γ)(1 + z) − γ]ηs + γS λη | {z } | {z s } vb (ηs ) κ(ηs ) λzηs2
Workshop in Economic Theory, UCLA
32
Gara Mínguez Afonso
Liquidity and Congestion
Market Frictions and Flow of Investors
¦ Rearranging, we get: µ γ
γ 1+ ληs
¶ (S − ηs ) =
a+b−1 X ³ j=a
´j a + b − 1´³ x λzηs2 j r + γ λzηs2 + [(r + γ)(1 + z) − γ]ηs + γS ³ ´a+b−1−j λzηs2 x 1− r + γ λzηs2 + [(r + γ)(1 + z) − γ]ηs + γS
¦ Polynomial of degree 2(a + b) in ηs.
Workshop in Economic Theory, UCLA
33
Gara Mínguez Afonso
Liquidity and Congestion
Simple Case a = 1, b = 1 ¦ Indifference condition: vb = κ x λzηs2 = κ + (κ − κ)γ r + γ λzηs2 + [(r + γ)(1 + z) − γ]ηs + γS
µ
γ 1+ ληs
¶ (S − ηs )
¦ Reorganising terms yields a polynomial of degree four in ηs : h i x 2 4 2 λ z(κ − κ)γηs + λ(κ − κ)γC − λzD + λ z ηs3 + r+γ h i 2 + λ(κ − κ)γ S(1 − z) − CD ηs2 − h i 2 − γSD + (κ − κ)γ SC ηs − (κ − κ)γ 3 S 2 = 0 where C = (r + γ)(1 + z) − γ and D = λκ + λ(κ − κ)γS − (κ − κ)γ 2 . ¦ There exists closed-form solution to this equation.
Workshop in Economic Theory, UCLA
34
Gara Mínguez Afonso
Liquidity and Congestion
Market Frictions and Flow of Investors
¦ We solve for ηs∗ using the bisection method.
¦ Once we compute ηs∗, we can derive g ∗: g ∗ = g ∗(λ, γ, |{z} r , |{z} S , |{z} x , |{z} z , |a, b,{zκ, κ}) −
Workshop in Economic Theory, UCLA
+
+
+
35
Gara Mínguez Afonso
Liquidity and Congestion
0.56
1 0.54 r=0.01 r=0.03 r=0.05 r=0.07
g*
g*
0.52
0.5
0.6
0.4
0.48
0.2
0.46
0.44 2
S=4 S=3 S=2 S=1
0.8
4
6
8
10
λ
12
14
16
18
0 2
20
0.65
4
6
8
10
λ
12
14
16
18
20
0.75 0.7
0.6
x=2 x=1.25 x=0.75 x=0.5
0.55
z=5 z=2 z=1 z=0.5
0.65
g*
g*
0.6 0.55
0.5
0.5 0.45 0.45 0.4 2
4
6
8
10
λ
12
14
16
18
20
0.4 2
4
6
8
10
λ
12
14
16
18
20
Figure 10: g ∗ as a function of λ for different values of r, S, x and z.
Workshop in Economic Theory, UCLA
36
Gara Mínguez Afonso
Liquidity and Congestion 20
1
18
0.8
16
g*
0.6
14
0.4
12
λ
0.2
10
0 20
8 15
6 10 5
λ
0
1
0
4
4
3
2
2
0.5
γ
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
1.5
2
γ
2.5
3
γ=0.5 γ=1 γ=1.5 γ=2 γ=2.5
g*
0.9
g*
0.9
1
0.4
0.4
0.3 0.2 0.1 0
0.3
γ=0.5 γ=0.4 γ=0.3 γ=0.2 γ=0.1 2
4
6
8
10
λ
12
14
16
18
0.2 0.1 20
0
2
4
6
8
10
λ
12
14
16
18
20
Figure 11: g ∗ as a function of market efficiency λ and liquidity shocks γ.
Workshop in Economic Theory, UCLA
37
Gara Mínguez Afonso
Liquidity and Congestion
Market Frictions, Liquidity and Welfare
¦ The impact of market frictions on liquidity is non-trivial. ¦ Interesting example: Fire-sale. ¦ Measure welfare by the weighted sum of investors’ expected utilities (weights given by measure ofZevery type of investor): κ W = |ηbvb + η0{zv0 + ηsvs} + κf (κ)dκ (16) Winside investors
Workshop in Economic Theory, UCLA
∗
|κ {z } Woutside investors
38
Gara Mínguez Afonso
Liquidity and Congestion
Welfare ¦ Measure of welfare: Z W = Winside
investors
+ Woutside
investors
κ
κf (κ)dκ
= ηb v b + η0 v 0 + ηs v s + κ
∗
where:
Winside
investors
=
d 1 S − kx r r+γ
γ S r
£
investors
=
£
(γ + ληs )z + (r + γ) + ηs (r + 2γ + ληs )z + (r + γ)
h
Woutside
¤
(r + γ + ληs )z + γ
i
"
a a 1 − Fbeta (κ∗ ; a + 1, b) = 1− a+b a+b
Workshop in Economic Theory, UCLA
¤
X ¡a + b¢
#
a+b
j=a+1
j
(κ∗ )j (1 − κ∗ )a+b−j
39
Gara Mínguez Afonso
Liquidity and Congestion
Example: Fire-sale Baseline
Fire-sale 2
b
1.5
η*
1
η*
b
s
0.5 2
3
4
5
λ
6
7
8
9
η*
1
η*
0 s
0.5 0 2
10
3
4
5
3
4
5
6
7
8
9
10
6
7
8
9
10
λ
1
s
1 0.9
η* / η*
0.8
b
η*b / η*s
b
1.5
0
0
0 1
η*
s
η*
η* , η* , η*
η*b, η*0, η*s
2
0.7
0.5
0.6 1
2
3
4
5
λ
6
7
8
9
10
0 2
λ
Figure 12: Equilibrium measures of investors and ratio of buyers-to-be to sellers-to-be as a function of efficiency λ. (γ = 0.4)
Workshop in Economic Theory, UCLA
40
Gara Mínguez Afonso
Liquidity and Congestion 128
η* η*
1
η*
s
0.5 3
4
5
6
λ
7
8
9
126
122
10
2
3
4
5
3
4
5
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
λ
2
v*b
η*b / η*s
s
124
1
0.5
0 2
0
v*
0
0
0 2
p* v*
s
b
1.5
p*, v* , v*
η*b, η*0, η*s
2
3
4
5
6
λ
7
8
9
1.5
1 2
10
78.5
λ
250 249.5
78
249 248.5
Welfare
Illiquidity
77.5
77
76.5
248 247.5 247 246.5
76 246
75.5 2
3
4
5
6
λ
7
8
9
10
245.5 2
λ
Figure 13: Measures, utilities, price, illiquidity and welfare as a function of λ.
Workshop in Economic Theory, UCLA
41
Gara Mínguez Afonso
Liquidity and Congestion
Example: Fire-sale
¦ Result 2 Diminishing market frictions can deteriorate market liquidity and reduce welfare.
¦ I NTUITION : Reducing frictions in this distressed market magnifies the effect of congestion and results in a higher price discount and hence in a less liquid market. Also, investors who hold this asset and those trying to sell it are clearly worse-off as the market becomes more one-sided, which leads to a decrease in overall welfare.
Workshop in Economic Theory, UCLA
42
Gara Mínguez Afonso
Liquidity and Congestion
Example: Fire-sale (a)
(b) 0.36
78
0.34
Trading Volume
78.5
Illiquidity
77.5
77
76.5
76
75.5 2
0.32
0.3
0.28
0.26
3
4
5
6
λ
7
8
9
10
0.24 2
3
4
5
6
λ
7
8
9
10
Figure 14: Illiquidity measured by price discount (a) and trading volume (b) as a function of the market efficiency λ.
Workshop in Economic Theory, UCLA
43
Gara Mínguez Afonso
Liquidity and Congestion
Example: Fire-sale
¦ Result 3 Market illiquidity, measured by the price discount, can increase while trading volume rises.
¦ I NTUITION : During a fire sale, reducing search frictions amplifies the effect of congestion, resulting in a less liquid market (Result 2).
But a more
efficient search process also increases the frequency of meetings between the investors. More frequent meetings translate into a higher trading volume.
Workshop in Economic Theory, UCLA
44
Gara Mínguez Afonso
Liquidity and Congestion
Related literature ¦ Self-fulfilling liquidity: Brunnermeier and Pedersen (2007), Gromb and Vayanos (2002), Pagano (1989), Dow (2004), Plantin (2004) ¦ Search models: Phelps (1972), Diamond (1982), Pissarides (1985) ¦ Search in asset pricing: – Duffie et al. (2005, 2007), Lagos and Rocheteau (2007) – Vayanos and Wang (2007), Weill (2007), Vayanos and Weill (2007) ¦ Asset pricing with exogenous trading costs: Amihud and Mendelson (1986), Vayanos (1998), Acharya and Pedersen (2005) ¦ Huang and Wang (2007)
Workshop in Economic Theory, UCLA
45
Gara Mínguez Afonso
Liquidity and Congestion
Conclusions ¦ Search-based model to study the impact of market frictions on liquidity and asset price. ¦ Market liquidity depends on a tradeoff between complementarities and a congestion effect. When congestion dominates: 1. Reducing frictions can decrease market attractiveness. 2. Diminishing market frictions can deteriorate market liquidity and reduce welfare. 3. Market illiquidity (measured by the price discount) can increase while trading volume rises.
Workshop in Economic Theory, UCLA
46
