OPTIMAL PORTFOLIO BALANCING UNDER CONVENTIONAL PREFERENCES AND TRANSACTION COSTS EXPLAINS THE EQUITY

PREMIUM PUZZLE ANU Trevor Swan Distinguished Lecture 23rd May 2006

Peter SwanUNSW

What is the Equity Premium Problem? 1896-1994 NYSE excess equity return over T-bill

rate, 6% pa (Campbell, Lo and MacKinlay) For last 50 years excess return 8% pa Mehra and Prescott (1985) attempt to account for it

within a representative investor framework Risk aversion explains less than 0.4% No frictions such as transaction costs Preferences, expectations, endowments identical Provides no/limited motivation for trading activity in equity

and bonds

Liquidity (tradability) is related to returns Money is the most liquid of all assets but

expected return is negative! Treasury securities (e.g. T-Bills) have yielded

a very low return (0-2% real) in last 100 years Equities have yielded 6-8% more and are 26

times less liquid than Treasury securities

Table I: Derivation of Turnover Rates for US Treasury Securities and NYSE Equities, 1980-2004. Bond turnover rate is 26 times equity turnover rate (column 8)

Year 1 2 3 4 5 6 7 8

1980 616 11.4 6.9 7.72 11,352 31,871 0.36 21.67

1981 683 13.3 11.2 9.32 11,854 36,004 0.33 28.32

1982 824 17.4 14.8 10.16 16,458 38,907 0.42 24.01

1983 1,024 23.3 18.8 10.69 21,590 42,317 0.51 20.94

1984 1,247 28.5 24.3 11.01 23,071 47,105 0.49 22.47

1985 1,438 39.6 35.8 13.64 27,511 50,759 0.54 25.16

1986 1,619 53.3 42.3 15.35 35,680 56,024 0.64 24.11

1987 1,725 64.6 45.6 16.61 47,801 65,711 0.73 22.84

1988 1,821 63.0 39.2 14.59 40,850 73,989 0.55 26.43

1989 1,945 69.8 43.1 15.09 41,699 79,574 0.52 28.79

1990 2,196 68.7 42.5 13.17 39,665 86,852 0.46 28.83

1991 2,472 78.5 49.0 13.41 45,266 95,177 0.48 28.20

1992 2,754 95.7 56.4 14.36 51,376 107,731 0.48 30.11

1993 2,990 107.7 65.9 15.10 66,923 123,446 0.54 27.85

1994 3,126 116.1 75.2 15.91 73,420 136,667 0.54 29.62

1995 3,307 112.7 80.5 15.19 87,218 148,500 0.59 25.86

1996 3,460 117.3 86.4 15.31 104,636 165,832 0.63 24.26

1997 3,457 120.9 91.2 15.95 133,312 192,017 0.69 22.98

1998 3,356 126.5 100.1 17.56 169,745 223,196 0.76 23.09

1999 3,281 101.3 85.3 14.79 203,914 260,116 0.78 18.86

2000 2,967 98.6 108.0 18.11 262,478 297,433 0.88 20.52

2001 2,968 138.8 159.1 26.10 307,509 327,723 0.94 27.82

2002 3,205 170.8 195.6 29.72 363,136 345,709 1.05 28.30

2003 3,575 200.8 232.7 31.53 352,398 354,784 0.99 31.74

2004 3,846 226.3 270.0 33.55 365,352 369,042 0.99 33.89

Average 16.56 0.64 25.87

Security Turnover Rates for Treasury Securities (U.S.) and Equities (NYSE), 1980-2004

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

Year

Tre

asu

ries

0

0.2

0.4

0.6

0.8

1

1.2

Eq

uit

ies

Treasury

NYSE

Why is the illiquidity premium not as high on bonds as it is on equity despite the high propensity to trade bonds? In the US Treasury Securities turnover (trade relative to the

stock) 16 times pa and equity, 0.64 times, 1980-2004. Difference of 26 times average over 25 years. Gilts in the UK and Treasury securities in Australia almost as liquid relative to equity.

Treasury Securities are not subject to asymmetric information so relative trading costs much lower, close to 3 basis points.

Any model of the equity premium and security trading must be able to explain the extreme liquidity of government securities relative to equity.

In my model investors, all with identical preferences, are indifferent between trading equity with and without trading costs and similarly bonds and adopt the same investment horizon for all types of investments.

What do I do?

I relax the representative investor model within a simple mean-variance (Constant Absolute Risk Aversion) framework

Allow for endowment differences in simple trading model of Pagano (1989) and apply methodology of Constantinides (1986) equates expected utilities of investors with different trading costs

Incorporate proportional trading costs at an exceedingly low level (0.5% one way cost for equity)

Calibrate model to replicate observed trading levels for equity and bonds with observed volatilities1899-1994

What do I find (expressed in more technical

terms)? Gains from trade motivated by portfolio rebalancing are high Small frictions such as trading costs dramatically reduce the gains

from trade An illiquidity premium of 6% or more in excess of bond rate arises

as compensation for illiquidity to make investors indifferent between the trading asset with and without the transaction cost

Illiquidity premium is in addition to any risk premium and does not require significant risk or consumption volatility Variance bounds tests are not applicable to the illiquidity

premium Calibration for observed security trading ensures that impact of

trading cost on asset price is significantly higher (appox. 180 fold) than without calibration as in Constantinides (1986) model

What do I find?

Alleged “irrational exuberance” of equity markets of Greenspan and Shiller does not reflect irrationality but rather changes in trading volume and gains from trade

Most other alleged anomalies in asset markets are also explained Why the cross sectional return on NYSE depends largely on

stock turnover Low bond yield puzzle 20% return on “equity letter stock” puzzle Why “on the run” bonds have a lower return than “off the run”

bonds Why there is a positive term structure

My model implies that asset prices depend largely on the benefits of trading as reflected in the elasticity of trading demand When I use my model to extract the implied elasticity of trading

demand from empirical excess equity return data I find it agrees with the independently estimated demand elasticity

My theoretical are consistent with numerous empirical findings Amihud and Mendelson (1986) find that small

trading costs can explain relatively small asset return differences but find substantial empirical effect

Datar, Naik, and Radcliffe (1998) find that stock turnover significantly explains the cross-section of asset returns after controlling for Fama-French factors

Amihud (2002) provides further evidence that “excess” equity return at least partly represents an illiquidity premium.

Easley, Hvidkjaer and O’Hara (2002), and Easley and O’Hara (2004) provide further extensions

Chen and Swan (2006) Explanation for China “A” and “B” Stock Premia “A” stock traded domestically, “B” internationally; Stocks otherwise

identical Shanghai: “A” stock turnover 3.8 times more rapidly than “B” stock;

relative return 0.248; 0.507 cheaper to trade; relative price 5.98 Shenzhen: “A” stock turnover 5.65 more rapidly; relative return 0.4;

0.52 cheaper to trade; relative price 5.14 International investors require higher relative return to compensate

for higher trading cost Domestic investors compensated by high liquidity My Liquidity Asset Pricing Model explains relative returns and

changes in relative prices in the “A” and “B” markets far better than firm size, book to market ratio, informativeness of order flow

Simple intuition behind model

Simple supply and demand model based on differences in endowments Would-be suppliers have “excessive” number of risky securities (shares) Would-be demanders have bonds but too few risky securities

Risk as measured by variance depends on the square of asset holdings in a simple mean-variance framework

Small departures from equality of holdings by the two types leads to considerable “deadweight” costs and trade motivation

Most of the cost of trading (15 to 1 where 1 is “actual costs”) is due to the loss from not making trades which should have been made but were not due to transactions costs

Bid-ask spread is a wedge between supplier and demander

Numerical simulation, 1889-1994

Equity Market

Return 6% pa

Variance = 3.24 pa

2 way Transactions Cost 0.098%

CARA Coeff b = 1

Seller Endowment 7.5 shares

Buyer Endowment 0

Number of Traders N =10,000

Bond (T-bill) Market

Return 2% pa

Variance = 0.36 pa

2 way Transactions Cost 0.03%

CARA Coeff b = 1

Seller Endowment 7.5 T-bills

Buyer Endowment 0

Number of Traders N=10,000

Figure 2: Six Percent Illiquidity Premium with Moderate Volatility, Base-Case Equity Simulation from Table II

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

0.00

0.74

1.48

2.23

2.97

3.71

4.45

5.20

5.94

6.68

7.42

8.17

8.91

9.65

10.3

911

.14

11.8

8

Annualized Stock Turnover Rate

Rou

nd-T

rip T

rans

actio

n C

osts

Transactions Cost

Transactions Demand Schedule

Dead-Weight Loss Area

Iliquidity Premium

Transactions Cost Area

Figure 3: Simulation of High Treasury Security Turnover Rate; Base-Case Treasury Simulation from Table II

0.00%

0.02%

0.04%

0.06%

0.08%

0.10%

0.12%

0.00

0.74

1.48

2.23

2.97

3.71

4.45

5.20

5.94

6.68

7.42

8.17

8.91

9.65

10.3

911

.14

11.8

8

Annualized Bond/T-Bill Turnover Rate

Rou

nd-T

rip T

rans

actio

ns C

osts

Transactions Cost

Bond Transactions Demand Schedule

Transactions Cost Outlay Rectangular Area

Dead-Weight Loss

Illiquidity Premium

Trading Benefit Triangular Area

Derivation of stock turnover demand Computing the change in stock from the stock demands yields:

Increasing market depth by doubling the number of investors rotates

the turnover function anti-clockwise around the autarky point:

1 02

1 2

2 1

D D

T T

K K N Ra h a

K N b K

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

0 1 2 3 4 5 6

Stock Turnover Rate (Annualized)

Tw

o-W

ay R

elat

ive

Tra

nsac

tions

Cos

ts

Number of Investors = 4

Number of Investors = 8

Autarky Point

More supporting evidence

Datar, Naik, and Radcliffe (1998) conclude that a 1% drop in the monthly % stock turnover rate for NYSE stocks increases the cross-sectional monthly return by 4.5 basis points over the period, 1962-1991, conditional on the Fama-French (1992) factors, size, book to market, and CAPM beta.

Evaluating the slope of the compensation function with respect to turnover in base case (Table II) and using constant elasticity specification

or four basis points a month, closely replicating Datar et al.

0.000377c

A test of the model based on “Letter Stock” returns Privately placed “Letter Stock”, which cannot be

traded for typically 2 years, sells for discount of 17.5 to 20% pa.

With moderate volatility and relative trading cost in constant elasticity model set to choke off demand, my model generates a 19.5% pa return.

Letter Stock puzzle illustrates the principle that observed “resource” trading costs (here zero) are irrelevant and maximum illiquidity premium occurs in autarky situation

Constant elasticity trading demand, as opposed to linear, model for empirical purposes

is relative transaction costs. Illiquidity premium is

1

1and , 1

mp

Ra

p

1

, as 0,1 1

c x dx

1

, 11

c c

Monthly returns on the Australian Stock Exchange, 1994-1998 A monthly “equity premium” database with 24,350

observations for 576 stocks Estimated two simultaneous equations using non-linear

OLS and “constant elasticity” version of illiquidity premium model (where rho is inverse of beta):

Implied T-bill transaction cost 0.000311b

(

0.78137; 1.2798 , t-stat. = 139.63),

1 1et et b t1c =

1et t t

Table III: Summary statistics for the sample of approximately 576 Australian securities listed on the Australian Stock Exchange, 1994-98, with 24,350 Monthly Observations.

Allowing separate estimates of the turnover elasticity from the two equations the implied turnover elasticity estimated from the first equation is = 0.798021 and from the second, = 0.760572.

Statistic Equity Premium, c Trans. Cost, Turnover, Market Cap.

Mean -0.061887 0.041707 0.27575 pa $726.4 m. Stand. Dev. 0.83756 0.039103 0.39685 0.261181010 Median -0.063089 0.029113 0.159 pa $68.1 m

Table IV: Regression results for the sample of 576 Australian securities listed on the ASX, 1994-98.Column 1 2 3 4 Item Full Sample Above

Median Market Cap.

Below Median Market Cap.

Separately Estimated Elasticities

Intrinsic Liquidity Coeff. ( ) )807.41(

01455.0 )217.31(

021293.0 )379.14(

009559.0 )266.24(

015847.0

Inverse of Turnover Elastic, )63.139(

2798.1 )199.86(

3861.1 )674.53(

1476.1 NA

Equity Inv of Tover Elastic e )396.85(

2531.1

Inverse of Turnover Elastic )412.76(

3148.1

Turnover Elasticity ( ) 78137.0 72145.0 87138.0 NA

Elasticity estimated from the

Equity Premium ( e )

NA NA NA 0.798021

Elastic est from Turnover ( ) NA NA NA 0.760572

Implied T-bill TCost ( b ) 0.000311 0.00027 0.000443 0.000279

Identity of the elasticity estimates from turnover and equity premium data

Estimated separately for 90 stocks:

Illiquidity premium model implies that the estimates of the demand elasticity will be the same from both equations

The estimate the same in both equations if = 0. Same for 70 of the 90 single equation estimates

1

et et t

0 11t et et tc

Conclusions

In this paper I put forward the first coherent explanation for the “illiquidy”, i.e., equity premium of 6 to 8% based on “low” equity (proportional) trading costs of less than 0.5% (one way) in the realistic context of substantial gains from trade

The explanation is consistent with virtually all the know facts about the returns on and trading of both equity and T-Bills over the period, 1889-1994

The model breaks new ground in a whole host of dimensions, simplicity, closed form solutions, tractable, intuitive and capable of explaining realistic levels of trading demand, unlike the representative investor model.

The model is capable of generating a whole host of new empirical tests, some of which I have presented today and explaining numerous “anomalies”.

APPENDIX-THEORETICAL MODEL

The model is exceedingly simple, conventional, transparent and with a “closed form” solution Investor preferences are constant relative risk

aversion (CARA) Payoff (gross) is from a normal distribution with

expectation N/2 investor pairs with demanders excessively

endowed with riskless bonds and suppliers with the risky asset

Each pair has assets Endowment heterogeneity is

E d

E E 2 Var , , 1,...,i i iu c c b c i i N

0 0T S DK K K

0 0S D

T

K Kh

K

Budget Constraints of Suppliers and Demanders

Supplier budget constraint:

depends on dividend plus gross return (R=1+r) on bonds generated from sale of shares

Demander budget constraint:

depends on dividends less loss of bond interest and dollar spread, a, on value of equity purchased.

1 1 0 0 1

S S S S S S Sc w dK R w p K K

1 1 0 1 0

D D D D S D Dc w dK R w p a K K

Conjectured linear demands that are consistent (but investors are price-takers in simulation)

Conjectured linear demand incorporating the spread a:

and supply:

Representative Demander computes his residual demand and optimises wrt number of shares purchased:

Add ith Demander to both sides of the equation

1 1

21 1 1

2 2 2 2 2D S D S SN N N N N

K K N p a

1S S SK p

1D D SK p a

1 1 1 1

2 2 21

2 2 2 2 2 2D D S T D S S D

i i

N N N N N NK K K K N p a K

Maximize expected utility

Compute price reaction to increase in demand

Maximise expected utility wrt incremental purchase:

Substitute Demanders and Suppliers demands into conjectural variation condition:

1

10

1

S

Di

dp

dK N

1 0 2

11

01

D D DiS D

iDi

E u R K KR p a b K

K N

221Var

D Dic K

0 0

12

22 1 2 1

1

D SS S

Di

RK RKN NR p a Rp

N NK

Rb

N

1 1 1

2

2 2 2D D S T

i

N N NK K K K

Derivation of Nash equilibrium market clearing price

Computing the slope and equating it to conjectural condition yields slope .

Asset demands found by substituting into linear conjectures

Solving for the Demand price by summing the two types of demand yields the Nash equilibrium

2

2

1

N R

N b

0

1 2

2

1 1

SDD D S

R p aK NK p a

N N b

01 2

2

1 1

S SS S S K N Rp

K pN N b

2

22 2 2

TS D T

D S

bKK a a

p p aR

2

22 2 2

TS D T

S

bKK a a

pR

Derivation of asset demands

Mid-point price = Present Value using riskless rate of the expected dividend – risk adjustment which depends on the number of risky shares to be held by the trading pair.

Does not appear to be a function of transactions costs at all but the ask price is lower by the half-spread and thus does depend on transaction costs.

But asset demands, on substituting for the market clearing price, do depend explicitly on transaction costs:

01 2

1 2

1 2 1

DD D TK N R

K f a K aN N b

01 2

1 2

1 2 1

SS S TK N R

K f a K aN N b

Market impact costs (enrich model but are not required or used in this application) These arise in “thin” markets because the strategic

investor, a large institution who is perhaps only one of a few large players in the market at that time, recognises his monopsony power; a large order will turn the terms of trade against himself.

Dampens trading, reduces ability to share risk, but does not have a large impact on equilibrium price.

Increases in risk aversion, volatility, shares on issue relative to trading pairs, and endowment heterogeneity all raise the optimal degree of mutual portfolio rebalancing.

How much compensation is required for loss of gains from trade for bearing transactions costs? Following Constantinides (1986), c(a) is the increase in expected per-period

dividend per share required to offset a $ spread cost of a leaving expected utility (of summed buyers and sellers) unaltered:

Equating expected utility of the trading pair with transactions costs to the pair without

Solving for the required increased rate of return (illiquidity premium) the compensation amount:

2 220 1 12

a T T T D SbU R w aK a c a K K K

02

20

2 1 1 2a T TRa N N R

U U U c a K K h aN N b

2

21

2 1 1 2 T

N RN Rc a a h a

N N b K

Equilibrium Price of Asset with Any Transaction Cost Equates Expected Utilities Across Equilibria

2

2T

mp S

bc a Kc a

P a pR R

S mpR p P a c a 2 Tb hK

a aR

2

2 2

1 1 2TN N h

c a b KN N

B a c a c a

mp mp c a c a B aP a P a

R R

In Constantinides (1986)/Swan Framework Asset Prices Fall When Transaction Costs Increase but Not in Conventional Framework True asset demand elasticity is always downward sloping in

transaction cost:

Conventional asset price elasticity with respect to transaction costs is positive when the elasticity of demand exceeds unity, indicating the peculiar result that untradeable assets are worth more!

10

1

mppa mp mp

c a a a aN

R p N p

11

mppa amp

a a

R p

What makes up the illiquidity premium? Made up of the actual cost of transacting plus the (“deadweight”) cost of not

being able to make the trades investors would have made in the absence of trading costs.

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

0.00

0.74

1.48

2.23

2.97

3.71

4.45

5.20

5.94

6.68

7.42

8.17

8.91

9.65

10.3

911

.14

11.8

8

Annualized Stock Turnover Rate

Rou

nd-T

rip T

rans

actio

n C

osts

Transactions Cost

Transactions Demand Schedule

Dead-Weight Loss Area

Iliquidity Premium

Transactions Cost Area

Constantinides’ formula: Annualised relative compensation for bearing transaction costs It diminishes in the length of the investment horizon, T (inverse of

frequency of shocks to portfolio).

In my model portfolio rebalancing due to endowment shocks (T) is two weeks and implicit in Constantinides (1986) is 20 years. His negligible compensation for bearing transaction costs arises in my model when negligible trading occurs, i.e., when the gains from trading are relatively low such that even small frictions eliminate trading almost entirely.

11 1

1

1 1annual mp

annual

mp mp mp mp

c a c a c a pe a

p p p p

Some unexpected comparative-static findings More compensation is required in ‘thicker’ markets because

trading gains are higher:

The increment to the illiquidity premium (slope of the illiquidity function wrt transaction costs) is higher the greater are the gains from trade:

Required compensation for bearing transaction costs equals the area under the stock turnover function from 0 to a (i.e., consumer surplus from transacting):

3

20

1

c a

N N

01

Nc a R a

N

01

a

x

RNc a x dx

N

Equity TbillZero Tcost Base Case Max Outlay Zero Tcost Base Case Max Outlay

1 2 3 4 5 6Annualised Return Variance, s 2 3.24% 3.24% 3.24% 0.36% 0.36% 0.36%Annualized Riskless Rate, r 2% 2% 2% 0% 0% 0%Coefficient of absolute risk aversion, b 1 1 1 1 1 1

Initial endowment of each seller, K0S

7.5 7.5 7.5 7.5 7.5 7.5

Initial endowment of each buyer, K0D

0 0 0 0 0 0Relative Endowment Heterogeneity, h 1 1 1 1 1 1Initial Bond Endow, Supplier, 0 0 0 0 0 0Initial Bond Endow, Demander 7.5 7.5 7.5 7.5 7.5 7.5Total Population of Traders, N 10,000 10,000 10,000 10,000 10,000 10,000Annualised Expected Net Return, 20.22% 20.22% 20.22% 2.06% 2.06% 2.06%Two sided prop dollar spread, a 0% 0.98% 0.506% 0% 0.03% 0.056%Prohibitive level of Trading Cost, abar 1.03% 1.03% 1.03% 0.11% 0.11% 0.11%TCost Prop of Mid-Point Price, a/pmp 0% 0.978% 0.505% 0% 0.030% 0.056%Equilibrium Demand by Seller, K1

S3.750 7.383 5.625 3.75 4.75 5.625

Equilibrium Demand by Buyer, K1D

3.750 0.117 1.875 3.75 2.75 1.875Equil Amnt Sold per Fortnight, K 3.750 0.117 1.875 3.75 2.75 1.875Annualised Turnover Rate, 12 0.38 6 12 8.80 6Annual Illiq Prem per unit Tcost, c(a)/a 11.980 6.177 8.985 12.000 10.400 9.000Annual Illiquidity Prem due Tcost, c(a) 0% 6.054% 4.545% 0% 0.312% 0.506%Ratio Unobserved to Observed Tcost NA 15.452 0.498 NA 0.182 0.500Utility of Trading Pair with TCosts NA 15.0694 15.0611 NA 15.0245 15.0243Utility of Trading Pair with No TCost 15.0694 NA NA 15.0245 NA NAAnnual Gains fm Trade rel Autarky, B(a) 6.075% 0.021% 1.530% 0.675% 0.363% 0.169%Equil Price without Tcost, pa=0 1.0025 1.0025 1.0025 1.0001 1.0001 1.0001Equil Mid-Point Price with Tcost, pmp 1.0025 1.0000 1.0006 1.0001 1.0000 0.9999Trading Elasticity wrt Tcosts, 0 -30.957 -1 0 -0.364 -1Elastic of Req Compen wrt Tcosts, 1.0025 0.0608 0.6684 1.0001 0.8462 0.6668Annual Slope Compn Fn wrt Tover, -0.243 -0.008 -0.122 -0.027 -0.020 -0.014

mppa

a

TK K

c

0Dw

0Sw

Ratio of illiquidity premium to transactions costs

This ratio, is 12 for small transaction charges and 6.2 in the base case. Constantinides (1986)

obtains an equivalent figure of 0.075 in a representative investor model providing little incentive to trade. His model is calibrated in many respects, but it is not calibrated for trading activity!

My result is about 180 times higher than this! Contantinides points out that my model also yields his result if

the investor horizon is 20 years instead of a fortnight! But negligible trading of equity and bonds occurs with a 20

year horizon (portfolio rebalancing shocks once every 20 years)! Less than 1% of what we observe.

Conclude that transactions costs only matter for asset prices if investors care about trading and gain sufficiently from trade.

1mp

e a c a

a p a