investment horizons and indeterminacy in financial markets shinichi hirota, juergen huber, thomas...

Investment Horizons and Indeterminacy in Financial Markets

Shinichi Hirota, Juergen Huber, Thomas Stoeckl, and Shyam Sunder

Theoretical and Experimental Macroeconomics, Barcelona GSE Summer Forum

June 11-12, 2015

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The purpose of this paper

• Explore – Why prices may deviate from fundamental

values in financial markets.

• Focus on– Investors’ short trading horizons and the

difficulty of backward induction and forming rational expectations.

• Conduct– Laboratory experiments

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Main Findings

• Prices tend to deviate from fundamental values (bubbles, indeterminacy) when investors have horizons shorter than the maturity of the securities they trade.

• Short-horizon investors fail to backward induct to others’ expectations of the second and higher orders to bring prices to the fundamental values.

• The shorter the investment horizon, (the larger number of generations), the more difficult the backward induction through higher orders, and the more likely that prices deviate from fundamentals.

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Background• Bubbles and price volatility in financial markets

are often attributed to short-term investors’ speculative trading.

• In standard finance theory, however, variations in decision horizons of investors do not enter the theory.

• Even in a market dominated by short-horizon investors, their backward induction is supposed to lead prices being close to the fundamental values.

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Critical assumption in finance theory

1. All generations of investors form rational expectations of future sales prices.

2. Rational expectation is common knowledge among all generations of investors

• By recursively forming REs, Pt = Ft is derivable as REE.

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In practice, backward induction may not occur

• Some generations of investors may not form rational expectations.

• Even if all generations of investors do, rational expectation may not be

common knowledge.

• Under such conditions, investors cannot backward induct from first and

higher order expectations to the present value of securities.

• Prices are no longer anchored to the fundamental values and become

indeterminate.

• We explore this possibility by conducting laboratory experiments.

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Fundamental Value vs. Price for a simple, single dividend security

Fundamental value:

Long-term Investor’s Valuation:

(1)

(2)

Short-term Investor’s Valuation:

)( mtttt DEVP

)( mttt DEF

)( ktttt PEVP (3)

Pt is not necessarily equal to Ft

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For Pt to be equal to Ft

• Rational Expectation of P t+k

• Homogeneous Investors

• The Law of Iterated Expectations　• By recursive process, Pt = Ft is derivable by

the backward induction.

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Difficulty of Backward Induction• Backward Induction may fail.

– Infinite maturity (rational bubbles) • Blanchard and Watson (1982), Tirole (1985)

– Infinite number of trading opportunities • Allen and Gorton (1993)

– Heterogeneous Information• Froot, Scharfstein, and Stein (1992), Allen, Morris, and Shin (2002)

– Rationality may not be common knowledge• Delong et al. (1990a)(1990b), Dow and Gorton (1994)

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Price Bubble sans Dividend Anchors

• There are cases where short-term investors have difficulty in backward induction.

• Stock prices (Pt ) form deviate

•

• from fundamentals ( Ft )

No longer anchored by future dividends

)( ktttt PEVP

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In an Earlier Experimental StudyHirota, Shinichi and Shyam Sunder. “Price Bubbles sans Dividend Anchors: Evidence

from Laboratory Stock Markets,” Journal of Economic Dynamics and Control 31, no. 6 (June 2007): 1875-1909.

• What happens when short-term investors have difficulty in the backward induction?

• Two kinds of the lab markets – (1) Long-term Horizon Session– (2) Short-term Horizon Session

• Bubbles (positive and negative) tend to arise in (2), but not in (1)

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Long-term Horizon Session

Single terminal dividend at the end of period 15.

An investor’s time horizon is equal to the security’s maturity.

Prediction: Pt = D

Period 1 Period 15

D(Trade)

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Short-term Horizon Session

Single terminal dividend at the end of period 30.

The session will “likely” be terminated earlier.

If terminated earlier, the stock is liquidated at the following period predicted price.

An investor’s time horizon is shorter than the maturity and it is difficult to backward induct.

Prediction: Pt D

Period 1 Period x Period 30

DEx (Px+1)(Trade)

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Figure 4: Stock Prices and Efficiency of Allocations for Session 4(Exogenous Terminal Payoff Session)

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Figure 5: Stock Prices and Efficiency of Allocations for Session 5(Exogenous Terminal Payoff Session)

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Figure 6: Stock Prices for Session 6 (Exogenous Terminal Payoff Session)

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Figure 7: Stock Prices and Efficiency of Allocations for Session 7(Exogenous Terminal Payoff Session)

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In long-horizon sessions

• Long-horizon Investors play a crucial role in assuring efficient pricing.– Their arbitrage brings prices to the fundamentals.

• Speculative trades do not seem to destabilize prices.– 39.0% of transactions were speculative trades.

• By contrast, in short horizon treatments:

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Figure 8: Stock Prices and Efficiency of Allocations for Session 1 (Endogenous Terminal Payoff Session)

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Figure 9: Stock Prices and Efficiency of Allocations for Session 2 (Endogenous Terminal Payoff Session)

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Figure 10: Stock Prices and Efficiency of Allocations for Session 8(Endogenous Terminal Payoff Session)

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Figure 11: Stock Prices and Efficiency of Allocations for Session 9(Endogenous Terminal Payoff Session)

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Figure 12: Stock Prices for Session 10(Endogenous Terminal Payoff Session)

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Figure 13: Stock Prices for Session 11 (Endogenous Terminal Payoff Session)

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Discussion (short-horizon sessions)

• Price levels and paths are indeterminate.– Level

• Small Bubble (Session 1)• Large Bubble (2, 8, 9, 10)• Negative Bubble (11)

– Path• Stable Bubble (1, 11, 2 ?)

– Rational Bubble• Growing Bubble (8, 9, 10)

– Amplification Mechanism, Positive Feedback

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Result

• In the long-horizon sessions, price expectations are consistent with backward induction.

• In the short-horizon sessions, price expectations are consistent with forward induction.

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However, Objections to Design of the Short-Horizon Sessions

Single terminal dividend at the end of period 30.

The session will “likely” be terminated earlier.

If terminated earlier, the stock is liquidated at the following period predicted price.

Environment not fully specified

In the current work, we use a fully specified overlapping generations structure

Period 1 Period x Period 30

DEx (Px+1)(Trade)

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Laboratory Experiment

• All markets have 16 periods of trading– Each period lasts for 120 seconds.

• Single kind of simple assets– Single, certain, common knowledge terminal dividend

of 50 at the end of period 16 (Fundamental value = 50).

• Control the length of trading horizon of investors– Overlapping generations structure (see the next slide).

• High / Low liquidity treatment– See the slide after next

Markets with Overlapping Generations of Traders

• Every period has two overlapping generations of five traders each in the market

• Only one initial generation is endowed with assets (single common knowledge dividend of 50 paid at maturity—end of period 16)

• All other generations enter with cash, can buy assets from the “old” generation, and sell them when they become “old” to exit the market with cash

• Individuals may re-enter after sitting out the market for one or more (random number) of generations (in T4 and T8 only)

• Each session is repeated six times (independently with different subjects)

• Equilibrium transaction volume per session: 160

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Overlapping Generations Experimental DesignTreat-ment

Period# ofSubjects

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 End of 16

T15 G05 G1 D

T25 G05 G15 G2 D

T4

5 G05 G15 G25 G35 G4 D

T8

5 G05 G15 G25 G35 G45 G55 G65 G75 G8 D

Notes: D means that the last generation of investors receives terminal dividends (50) at the end of Period 16.

Table 2: Treatment Overview

    Liquidity

    Low (C/A ratio=2)

High(C/A ratio=10)

Number of

entering

generations

1 T1L T1H

2 T2L T2H

4 T4L T4H

8 T8L T8H

Table 3: Treatment ParametersTreatment T1L T1H T2L T2H T4L T4H T8L T8HMarket setup                No. of generations 2 2 3 3 5 5 9 9Terminal dividend 50 50 50 50 50 50 50 50Initial No. assets/trader G0 32 32 16 16 8 8 4 4Initial No. assets G(i) 0 0 0 0 0 0 0 0Total assets outstanding 160 160 80 80 40 40 20 20Total value of assets 8,000 8,000 4,000 4,000 2,000 2,000 1,000 1,000Initial cash/trader G0 0 0 0 0 0 0 0 0Initial cash/trader G(i) 3,200 16,000 1,600 8,000 800 4,000 400 2,000Total cash 16,000 80,000 8,000 40,000 4,000 20,000 2,000 10,000Cash-asset-ratio (C/A-ratio) 2 10 2 10 2 10 2 10Invited subj. (3n+3) 15a 15a 18 18 18 18 18 18Participating subjects 90 90 108 108 108 108 108 108       Exchange rates (Taler/€)      Generation 0 (G0) 100 100 100 100 100 100 100 100Transition generations   100 500 100 500 100 500Last generation 200 1,000 200 1,000 200 1,000 200 1,000Predictors 133 133 133 133 133 133 133 133Exp. payout/subject (euros) 16 16 16 16 16 16 16 16

NOTES: The following parameters are identical across all treatments: Number of traders/generation (5); number of active generations (2); market size (10 traders); period length (120 sec.); total number of periods (16); number of markets per treatment (6); number of expected transactions (160).a In treatments T1LH we invited 15 subjects instead of 18 as no subject pool for future generations is needed. However we invited five subjects to serve as predictors.

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Continuous double auction markets

Trader: Information about your task (trader), period you leave the market, current Share and Taler holdings.Predictors: Information about your task (predictior) and your forecast.

Current Market Price (of Stock)

Price-Chart of current period

SELL: You sell one unit, given the price with the blue background.

BUY: You buy one unit, given the price with the blue background.

List of all ASKS: from all traders - your own asks are written in blue. The ask with blue background is always the most attractive, because it is the cheapest for the buyer.

List of all BIDS: from all traders - your own bids are written in blue. The bid with blue background is always the most attractive, yielding the highest revenues for the seller.

ASK: seller’s analogue to BID - see above.

BID: enter the price you are willing to pay for one unit. Trade does not take place until another participant accepts your bid!!!

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Conducted Experiments

• Innsbruck-EconLab at University of Innsbruck

• September, October and November 2013

• A total of 828 University of Innsbruck students (bachelor and master from different fields).

Hypotheses I• Hypothesis I0 (REE):

Deviations of prices from the fundamental value do not differ during periods when only short-horizon traders are present compared to periods when long-horizon traders are present in the market.

• Hypothesis IA: Deviations of prices from the fundamental value are larger during periods when only short-horizon traders are present compared to periods when long-horizon traders are present in the market. 35

Hypotheses II

• Hypothesis II0 (REE):For a security of a given maturity, the deviation of prices from the fundamental value is not affected by the length of investors’ trading horizons.

• Hypothesis IIA : For a security of a given maturity, the deviation of prices from the fundamental value increases as the length of investors’ trading horizons becomes shorter.

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Experimental Results

Figure A3: Individual market results for T1L.

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G0

G1

40

G0

G1

G2

41

G0

G1

G2

G3

G4

42

G0

G1

G2

G3

G4

G5

G6

G7

G8

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Summary of results (low Liquidity)

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G0

G1

45

G0

G1

G2

46

G0

G1

G2

G3

G4

47

G0

G1

G2

G3

G4

G5

G6

G7

G8

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Summary of results (high Liquidity)

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Analyses of the Price Deviations

• Measure of mis-pricing

• Relative Absolute Deviation (RAD): Stockl, Huber and Kirchler (2010)

RAD =

• Period-RAD =

=

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Table 3: Average Period-RAD for each period

Panel A: High-liquidity session

Panel B: Low-liquidity session

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

T1 1.423 0.582 0.354 0.293 0.329 0.301 0.321 0.390 0.374 0.382 0.396 0.303 0.323 0.286 0.387 0.259

T2 1.825 1.016 0.310 0.406 0.467 0.536 0.541 0.477 0.676 0.865 0.705 0.313 0.232 0.468 0.232 0.179

T4 1.552 1.471 1.342 1.038 1.182 0.960 0.798 0.499 0.697 0.509 0.470 0.559 0.325 0.210 0.167 0.040

T8 1.879 1.249 1.373 1.392 1.409 1.498 1.177 0.991 1.108 1.082 1.607 1.733 1.019 0.647 0.550 0.273

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

T1 1.423 0.582 0.354 0.293 0.329 0.301 0.321 0.390 0.374 0.382 0.396 0.303 0.323 0.286 0.387 0.259

T2 1.825 1.016 0.310 0.406 0.467 0.536 0.541 0.477 0.676 0.865 0.705 0.313 0.232 0.468 0.232 0.179

T4 1.552 1.471 1.342 1.038 1.182 0.960 0.798 0.499 0.697 0.509 0.470 0.559 0.325 0.210 0.167 0.040

T8 1.879 1.249 1.373 1.392 1.409 1.498 1.177 0.991 1.108 1.082 1.607 1.733 1.019 0.647 0.550 0.273

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• This supports Hypothesis IA, rather than I0.• REE does not hold in our laboratory

Table 4: Comparison of average Period-RAD between periods with long-horizon investors and periods with only short-horizon investors

(1) Periods with long-horizon

investors

(2) Periods with only short-

horizon investors

Difference

(2)-(1)

High-liquidity session (H)

0.401

(177)

1.024

(204)0.623***

Low-liquidity session (L)

0.140

(178)

0.502

(203)0.362***

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Short-horizon investors have difficulty in forming RE even if it involves one future generation left.

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• This supports Hypothesis IIA, rather than II0.

• The shorter the investment horizons, the more price deviation from the fundamentals.

Table 5: Investment horizons and average Period-RAD

Treatment

(Average investment

horizon)

T1

(16.0 periods)

T2

(10.7 periods)

T4

(6.4 periods)

T8

(3.6 periods)

High-liquidity session (H)

0.421

(95)

0.586

(94)

0.739

(96)

1.187

(96)

Low-liquidity session (L)

0.116

(94)

0.355

(96)

0.429

(96)

0.429

(95)

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Liquidity Supply and Mispricing

• Prices tend to be above fundamental value in high-liquidity sessions, but below the fundamental value in the low liquidity sessions.

consistent with previous experiments

• Period-RD =

• The average of Period-RD = 0.534 in H, -0.222 in L• This liquidity effect on prices is larger when

there are only short-horizon investors in the market.

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• With high liquidity, short-horizon investors magnify the overpricing (bubbles).

• With shortage of liquidity, short-horizon investors magnify the undervaluation (liquidity crisis).

Table 6: Comparison of average Period-RD between periods with long-horizon investors and periods with only short-horizon investors

(1) Periods with long-horizon

investors

(2) Periods with only short-horizon investors

Difference

(2)-(1)

High-liquidity session (H)

0.295

(177)

0.741

(204) 0.446***

Low-liquidity session (L)

-0.087

(178)

-0.340

(203)-0.253***

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Price Expectations• Short-horizon investors have difficulty in

forming RE of future prices. Then, how do they expect future prices?

• The fundamental model

• The trend model

• The combined model

)()( tttktt PFPPE

)()( ktttktt PPPPE

)()()( ktttttktt PPPFPPE

Price Predictions/Expectations

• Hirota and Sunder (2007): results show that when subjects cannot do backward induction, they resort to forward induction, and simply project past data in forming their expectations about the future

• In long-horizon sessions, future price expectations are formed by fundamentals.

– Speculation stabilizes prices.

• In short-term sessions, future price expectations are formed by their own or actual prices.

– Speculation may destabilize prices.

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High-liquidity Session

Periods with long-horizon investors Periods with short-horizon investors

FUND TREND COMBINED FUND TREND COMBINED

Const. 1.672** -0.709 1.733** 4.159** -2.611* 0.515

(0.622) (1.595) (0.620) (1.895) (1.310) (1.449)

(Ft - Pt) 0.197*** 0.211*** 0.109* 0.078

(0.043) (0.053) (0.061) (0.057)

(Pt - Pt-1) 0.020 0.067 -0.301*** -0.270***

(0.031) (0.044) (0.049) (0.043)

N 173 167 167 186 168 168

F 20.96 0.42 8.09 3.19 37.71 25.16

p 0.000 0.522 0.002 0.092 0.000 0.000

adj. R2 0.38 0.00 0.39 0.14 0.30 0.36

Table 7: Price expectations model estimates

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Low-liquidity Session

Periods with long-horizon investors Periods with short-horizon investors

FUND TREND COMBINED FUND TREND COMBINED

Const. -2.275*** 1.054 -2.543** -0.804 -0.248 -1.636**

(0.684) (0.737) (0.742) (0.524) (0.399) (0.671)

(Ft - Pt) 0.401*** 0.419*** 0.070*** 0.065**

(0.092) (0.096) (0.017) (0.024)

(Pt - Pt-1) -0.088 -0.016 -0.162* -0.180**

(0.079) (0.031) (0.081) (0.074)

N 171 162 162 186 168 168

F 19.36 1.26 10.10 16.25 3.95 5.82

p 0.000 0.274 0.001 0.001 0.063 0.012

adj. R2 0.43 0.01 0.43 0.08 0.08 0.13

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Conclusions• Our experimental results showed that investors’ short

trading horizons cause price indeterminacy in financial markets.– Deviations of prices from fundamental values increase

significantly when only short-horizon investors are present in the market.

– Prices are more likely to depart from the fundamentals as investment horizons shrink.

– Price expectations are formed not based on fundamentals but based on recent price changes.

– The short investment horizons create upward pressure on prices when liquidity is high and downward pressure when liquidity is low.

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Implications• Frequently observed price bubbles (e.g., stock, gold,

real estate) may arise from short trading horizons of investors.

• The excess price volatility in real stock markets may be caused by the existence of short-horizon investors.

• Market inefficiency, anomalies, and behavioral phenomena more likely to be observed in markets dominated by short-horizon investors .

• The results doubt on the validity of investors’ ability to form RE - a commonly applied assumption in finance literature.

Thank You!

Shyam.sunder@yale.eduhttp://faculty.som.yale.edu/shyamsunder/research.html