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Speculation and Risk Sharing with New Financial Assets

Alp Simsek

Harvard University

May 2012

Alp Simsek (Harvard University) Speculation and Risk Sharing May 2012 1 / 22

Does �nancial innovation reduce risks?

Traditional view: Financial innovation helps diversify and share risks.

But new assets generate new uncertainties.Belief disagreements and speculation come as by-product. Tendsto increase risks.

Example from recent crisis: Subprime CDOs and their CDSs.

This research: E¤ect of �nancial innovation on portfolio risks whentraders have both risk sharing and speculation motives.


Consider a risk sharing setting with belief disagreements

Standard risk sharing model with mean-variance preferences:

Background risks and �nancial assets.

New assumption: Belief disagreements about asset payo¤s.

Financial innovation = Expansion of assets.

Measure of portfolio risks:

Average variance = Uninsurable variance + Speculative variance.

Main result: Financial innovation always decreases uninsurable varianceand always increases speculative variance.


Important mechanism: Hedge-more/bet-more

Speculative variance increases through two channels:

1 New assets generate new disagreements.2 New assets amplify speculation on existing disagreements(hedge-more/bet-more).

New asset on which there is agreement increases speculative variance!


Is risk sharing a major factor in endogenous innovation?

Endogenous �nancial innovation: Both risk sharing and speculationmotives for trade generate innovation incentives.

1 With common beliefs, endogenous assets minimize the averagevariance among all choices.

2 With large belief disagreements, endogenous assets maximize theaverage variance among all choices.

Belief disagreements change the nature of endogenous �nancial innovation!

Calibration for national income markets of Ath-Shiller (AER, 2001):Small belief disagreements su¢ cient for speculation to swamp risk sharing.


Outline of the talk

A simple example to illustrate the two channels.

The main result.

Brief discussion of welfare implications.


Consider a standard risk sharing setting

One consumption good (a dollar), two dates, f0; 1g.Trader, i 2 I , has:

Endowment, e, at date 0.Background risks: Random endowment, wi , at date 1.

Consumes only at date 1. Investment options:

Cash: Yields one dollar for dollar.Risky assets, j 2 J, in �xed supply (zero).

Asset j pays aj dollars at date 1, and trades at price pj date 0.


Consider mean-variance preferences with het. priors

Trader i solves:

maxxi

Ei [ni ]��i2vari [ni ] ,

s.t. ni = e � x0ip| {z }investment in cash

+ wi + x0ia.

Key assumption: Traders have heterogeneous prior beliefs.

Equilibrium: Prices, p, and allocations, fxigi , such that traders optimizeand markets clear (

Pi xji = 0 for each j).

Next: Simple example.


Example: Traditional common-beliefs benchmark

Risks, v1; v2 � N (0; 1), i.i.d. De�ne v = v1 + �v2.Two traders with �1 = �2 � � and risks w1 = v and w2 = �v .

Benchmark: Common (and correct) beliefs.

Autarky: Net worths, n1 = e + v and n2 = e � v .Innovation: Asset with payo¤ a1 = v introduced.Trader 1�s position and net worth:

x11 = �1 and n1 = n2 = e.

With common beliefs, �nancial innovation reduces portfolio risks.


Channel 1: New assets generate new disagreements

Next consider the case with belief disagreements:

Traders�beliefs for v2 is as before. Belief for v1:

Trader 1: N ("; 1) . Trader 2: N (�"; 1) :

Parameter, ", captures the level of disagreement.

Trader 1�s position and net worth:

x11 = �1|{z}risk sharing portfolio

+"

�

11+ �2| {z }

speculative portfolio

,

n1 = e +"

�

v1 + �v21+ �2

.

Large belief disagreements, " > ��1+ �2

�: Financial innovation

increases risks.


Channel 2: New assets amplify existing disagreements

Next consider the introduction of a second asset:

Earlier single asset case:

x11 = �1+"

�

11+ �2| {z }dampening

.

Additional asset with payo¤, a2 = v2 (agreement on payo¤).Trader 1�s position and net worth:

x11 = �1 +"

�|{z}asset 1, betting

and x21 = ��"�| {z }

asset 2, hedging

n1 = e +"

�v1.

New asset with agreement increases portfolio risks.

Intuition: Hedge-more/bet-more (and risk more).Alp Simsek (Harvard University) Speculation and Risk Sharing May 2012 11 / 22

General environment

Risks: v = (v1; ::; vm)0.

Net worths, wi , and asset payo¤s, aj , are linear combinations of v.

Assumption (A1). Traders�beliefs are given by fN (�vi ;�v)gi .=) They agree on variance of v. Might disagree on means.

Su¢ cient statistics:

N (�i ;�): trader�s belief for asset payo¤s.

�i : common belief for covariance of wi and a.


Characterization of equilibrium

Equilibrium portfolios: xi = xRi + xSi where:

xRi = ��1~�i| {z }Risk sharing portfolio

and xSi = ��1~�i�i| {z }

Speculative portfolio

,

Relative covariance: ~�i = �i ��

�i

1jI jX{2I�{,

Relative optimism : ~�i = �i �1jI jX{2I

��

�{�{.


Average variance and its decomposition

De�ne average variance of net worths:

=1jI jXi2I

�i��vari (ni ) .

Lemma: With common beliefs, the equilibrium portfolios minimize subject to resource constraints,

Pi xi = 0.

De�ne uninsurable variance, R , as the minimum possible .

De�ne speculative variance as the residual:

= R|{z}uninsurable variance

+ S|{z}speculative variance

.


Main result: Financial innovation increases spec. variance

Notation: Economy E�J�with assets J � J.

Compare E (JO ) and E (JO [ JN ) (old and new assets).

Theorem (Financial Innovation and Portfolio Risks)

(i) Financial innovation always reduces the uninsurable variance:

R (JO [ JN ) � R (JO ) .

(ii) Financial innovation always increases the speculative variance:

S (JO [ JN ) � S (JO ) .


Intuition for the main result

Consider the economy with only speculation reason for trade.

Then, a textbook result applies:

�Si|{z}std. of speculative portfolio return

=1�ie|{z}�relativei

SharpeSi| {z }Speculator�s Sharpe ratio

.

With more assets, SharpeSi increases through the two channels.

Thus, the speculative variance also increases.

Caveat: Equilibrium is Pareto e¢ cient. Should we be worried?


Based on a true story of famous economists


They decide to take a side bet, at some cost


They realize that this is Pareto optimal


The end of the story is unknown

The increase in portfolio risks ~Destroyed pillow.


Alternative welfare criterion: Belief-neutral ine¢ ciency

Suppose belief di¤erences come from behavioral distortions.

Practical problem: Which belief to use for welfare?

Solution by Brunnermeier, Simsek, Xiong (2012): Use all beliefs!

An allocation is belief-neutral ine¢ cient if it is ine¢ cient under anyconvex combination of agents�beliefs.

Financial innovation is belief-neutral ine¢ cient when it increases .


Conclusion

E¤ect of �nancial innovation on portfolio risks when traders have risksharing and speculation motive for trade.

Main result: Financial innovation increases speculative variance.Important channel: Hedge-more/bet-more.Endogenous innovation: Driven in part by belief disagreements.

Future work: Welfare e¤ects and policy implications.


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