April 10, 2003

Notes: More Finance Demand Curves Sloping the Wrong Way

Notes: Teaching: Econ 236: Behavioral: Finance: Yet More Demand Curves That Slope the Wrong Way

This time it's due to performance-based arbitrage: PBA: circumstances in which the fact that prices move against fundamentals leads investors to think that their smart-money managers aren't so smart, and so withdraw funds:

From Andrei Shleifer and Robert Vishny (1995), "The Limits of Arbitrage" (Cambridge: NBER Working Paper 5167).

One model of risky arbitrage is that of a large number of investors taking small positions against the mispricing. Fama's (1965) classic analysis of efficient markets and Ross's (1976) Arbitrage Pricing Theory are based on this model. An alternative, and in many cases more realistic view, is that arbitrage is conducted by a few professional, highly-specialized investors who combine their knowledge with resources of outside investors to take large positions. They operate in markets where fundamentals are difficult to ascertain and correct hedging strategies are hard to implement, such as the currency and derivative markets. The fundamental feature of such arbitrage is that brains and resources are separated by an agency relationship. the money comes from wealthy individuals, banks, endowments, and other investors with only a limited knowledge of individual markets, and is invested by arbitrageurs with highly specialized knowledge of these markets. In this paper, we examine such arbitrage and its effectiveness in achieving market efficiency.

The principal question we are interested in is: how effective is professional arbitrage in extreme circumstances, when prices are far away from their fundamental values? Intuition suggests that the greater is the deviation of prices from fundamentals in the absence of arbitrage, the greater the positions that arbitrageurs would take to counter these deviations, and hence the more effective is arbitrage. Put simply, greater mispricing calls forth more arbitrage resources. Indeed, the available models of risky arbitrage, such as Grossman and Miller (1988), DeLong, Shleifer, Summers, and Waldmann (1991), or Campbell and Kyle (1992), all share this feature. In these models, arbitrage is not perfect because arbitrageurs face either fundamental or noise trader risk (the risk that the mispricing will become even more extreme), Nonetheless, this risk does not outweigh the higher expected returns when mispricing is greater. At a substantive level, the implication is that arbitrage is stabilizing, and that it is more stabilizing in extreme circumstances.

In the analysis that follows, we analyze this logic in the agency context of arbitrage activity. The separation of knowledge and resources has three crucial implications for arbitrage activity. First, because outside investors are ignorant about the markets that arbitraguers invest in and cannot tell good arbitrageurs from bad, the resources they supply to the arbitrage activity are limited. Arbitrageurs can perhaps borrow funds in addition to resources they have under management, but their borrowing capacity is limited as well. Second, again because they are poorly informed, outside investors rationally use past performance of the arbitrageurs to gauge their ability and to increase and reduce funds they give them to manage. A good track record brings in more funds, and a bad track record causes a withdrawal of funds. Third, because arbitrageurs' knowledge is highly specialized, arbitrage markets are segmented. Only a relatively few experts, with good track records, can attract outside funds to engage in arbitrage in a given market.

These simple implications of agency have important consequences for arbitrage in extreme circumstances, when prices diverge significantly from fundamental values. Such price divergences can occur, for example, because unsophisticated investors experience a sharp move in their sentiment, and as a result try to significantly change their holdings. They may become highly enthusiastic about particular securities, or they may panic. In this paper, we focus not on the origins of unpredictable changes in investor sentiment, but rather on arbitrageurs' response. We show that, in the agency context of specialized arbitrage using outside funds, arbitrage can be very ineffective in returning prices to fundamental values when investor sentiment has driven them far away...

The Structure of the Game in "Limits to Arbitrage"

Model Building:

Three groups of agents: arbitrageurs, investors in hedge funds, and noise traders. Noise traders noisy. Arbitrageurs risk neutral.

Three periods: 1, 2, 3.

Fundamental value of asset is V. Arbitrageurs know this. Investors don't.

At time 3, value V becomes known to noise traders. Hence period-3 price is V.

At times 1 and 2, price is p(1), p(2).

In each of periods 1 and 2 noise traders receive a shock S(t) which generates an aggregate noise trader demand for the asset of:

QN(t) = [V - S(t)]/p(t)

Assume S(t) &Mac179; 0. Assume with probability q, S(2) = S > S(1). With probability 1-q, S(2) = 0 and p(2)=V.

In period 1, arbitrageurs know S(1). But they don't know S(2). They face noise trader risk in their positions from period 1 to period 2, and they cannot necessarily dynamically hedge by taking bigger positions in period 2 if they have lost due to noise trader risk between period 1 and period 2.

Arbitrageurs have resources under management of F(t). F(1) exogenous. F(2) as follows...

Consider only pessimistic noise shocks. If in period 2, S(2) = 0 and p(2) = V, then arbitrageurs invest in cash. If S(2) > 0 and p(2) < V, arbitrageurs want to invest all of F(2) in the underpriced asset. Their demand QA(2) = F(2)/p(2). Assume F(2) < S(2).

Asset in unit supply. Price then given by:

p(2) = V - S(2) + F(2)

Arbitrageur period 1 demand D(1)/p(1) may not be as large as F(1). In period 1:

p(1) = V - S(1) + D(1). Assume F(1) < S(1).

Now let's think about F(2). Competition between arbitrageurs. Investors in hedge funds are risk neutral. Investors are Bayesians. Different investors have different prior beliefs about arbitrageurs. No common knowledge. The market share of each arbitrageur is just the share of investors who believe he or she has the highest expected return.

Updating. "The underlying structural modle is sufficiently non-stationary and high-dimensional that investors are unable to infer the underlying structure of the model from past returns data. As a result, they only use simple updating rules about future returns based only on their priors and on any observation of past returns.

Hence, performance based arbitrage:

F(2) = G(D(1)*(p(2)/p(1) + (F(1)-D(1)), with G a function, G'>1, G''<0

Past performance of arbitrageurs determines the resources they get to manage: PBA. PBA critical.

Arbitrageurs are paid fees equal to marginal cost per dollar of funds under management. These funds are:

W = (V/p(2))*G(D(1)*(p(2)/p(1)) + F(1) - D(1))

Assume with probability q, S(2) = S > S(1). With probability 1-q, S(2) = 0 and p(2)=V.

When S(2)=0, arbitrageurs liquidate their position at a gain at t=2, and hold cash until t=3. W = G(D(1)*(V/p(1)) + F(1) - D(1))

When S(2) = S, arbitrageurs maximize the expected value of:

W = (1-q)(G(D(1)*(V/p(1)) + F(1) - D(1))) + q((V/p(2))*G(D(1)*(p(2)/p(1)) + F(1) - D(1)))

FOC: (1-q)(G'(D(1)*(V/p(1)) + F(1) - D(1))(V/p(1) - 1)) + q((V/p(2))G'(D(1)*(p(2)/p(1)) + F(1) - D(1)))((p(2)/p(1)) - 1) = 0

Look at the case in which D(1) = F(1), which happens when:

(1-q)G'(F(1)V/p(1))((V/p(1)) - 1 ) + qG'(F(1)p(2)/p(1))((p(2)/p(1)) - 1)(V/p(2)) > 0

The first term is the incremental benefit to arbitrageurs if they are fully invested from an extra unit of funding if the market recovers in period 2. The second term is the incremental cost of being fully invested when things get worse:

When arbitrageurs are fully invested in period 1, the second price solution will satisfy:

p(2) = V - S + G(F(1)p(2)/p(1))

If G' is not too high--if (G')(F(1)/p(1)) < 1--then there is a stable equilibrium. Otherwise, when prices fall at t=2, funds under management collapse and p(2) = V - S

The sensitivity of p(2) to noise trader risk:

d(p(2))/dS = -1/(1-G'F(1)/p(1) < -1 in stable equilibrium: p(2) falls more than one-for-one with the second-period noise-trader shock. In the extreme situation where noise trader shocks deepen starting from an already bad situation, arbitrageurs end up *reducing* their demand. When things get worse, arbitrageurs withdraw. Precisely when prices are furthest from fundamentals, arbitrageurs take the smallest position...

Look at Shliefer and Vishny (1992), Kiyotaki and Moore (1994), Stein (1995). Assets liquidated involuntarily at a time when the best potential buyers have limited funds and no external sources of capital. Fire Sale.


Andrei Shleifer and Robert Vishny (1992), "Liquidation Values and Debt Capacity: A Market Equilibrium Approach," Journal of Finance 47, 1343-66.

Jeremy Stein (1995), "Prices and Trading Volume in the Housing Market: A Model with Downpayment Effects," Quarterly Journal of Economics 110, 379-406.

Nobuhiro Kiyotaki and John Moore (1994), "Credit Cycles" (London: LSE).

Posted by DeLong at April 10, 2003 03:49 PM | TrackBack

Comments


The problem originates and is anchored in another
area which you have failed to address. You
are describing a situation in which no one
has done their homework.

pm

Posted by: P Morse on April 21, 2003 08:36 PM

The problem originates and is anchored in another
area which you have failed to address. You
are describing a situation in which no one
has done their homework.

pm

Posted by: Peter Morse on April 21, 2003 08:37 PM

The problem originates and is anchored in another
area which you have failed to address. You
are describing a situation in which no one
has done their homework.

pm

Posted by: Peter Morse on April 21, 2003 08:37 PM

So fund managers who cry wolf don't get more cash to play with?

I imagine some LTCM and Quantum fund traders can identify with this scenario quite well but it looks like a straw man to me.

By design this model provides the rest of the market with no way of making the "right" decision it doesn't get any information. The rational behaviour for the hedge fund manager is to make the investment and then share its information with the market. If the information is not enough to nail the price to V is it really information?

I am also unclear about the price being right at period 3. What is the right price at period 4?

It does demonstrate positive feed back but it is not clear to me in what sense the demand curve was not in the right direction except by assumption.

Posted by: Jack on April 22, 2003 07:58 AM

A further thought. This is really about views on past performance and mostly shows that people who follow trends follow trends. If the manager is good they will have a portfolio of trades and the majority of them will work and the money will not be withdrawn and may even be attracted and the manager if confident will put even more money into the trade!

Posted by: Jack on April 22, 2003 08:22 AM

any talk of fundamental values ignores the fact that very little agreement ever exists on what those values are, not to mention that they change almost as often as prices. and a significant number of market participants rely upon historical prices in making their decisions, making 'value' even harder to define in the traditional sense.

Posted by: selfish self promoter on April 27, 2003 05:32 PM

any talk of fundamental values ignores the fact that very little agreement ever exists on what those values are, not to mention that they change almost as often as prices. and a significant number of market participants rely upon historical prices in making their decisions, making 'value' even harder to define in the traditional sense.

Posted by: selfish self promoter on April 27, 2003 05:34 PM

any talk of fundamental values ignores the fact that very little agreement ever exists on what those values are, not to mention that they change almost as often as prices. and a significant number of market participants rely upon historical prices in making their decisions, making 'value' even harder to define in the traditional sense.

Posted by: selfish self promoter on April 27, 2003 05:34 PM
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