Limits to Arbitrage

The efficient market hypothesis states that prices are equal to their fundamental value. All deviations will be undone by rational arbitrageurs.

  • In theory, arbitrage is done by an infinite number of small risk-neutral investors. This leads to a direct price adjustment.
  • In reality, arbitrage is done by a small number of highly specialised risk averse institutional investors. This results in a slower price adjustment. This increases risks and costs.

This lesson will elaborate more on the topics related to limits to arbitrage.

Reasons for limits to arbitrage

There are mainly three reasons for limits to arbitrage (elaborated on in an article by Shleifer & Vishny, 1997).

  • Implementation costs
    The bid-ask spread could be too small if you take the transaction costs into account. Liquidity might also a problem. If the liquidity is too small, the buyer or seller need to buy/sell with slippage in order to fill the order. In other words, the buy/sell order could not be completed at the most optimal price. In cases like short selling or investing in derivatives, capital is need for margin requirements. These costs can cause a range around the fundamental price in which arbitrageurs do not find it worthwhile to trade against mis-pricing.
  • Noise trader risk
    Whatever caused the mis-pricing could make the pricing even worse. This could be bad for arbitrageurs on the short run, who could lose their position due to a margin call (for example). These type of market participant who cause the mis-pricing are named not to be full rational.
  • Fundamental risk
    The market can simply move against the position of the arbitrageur. Therefore, traders try to hedge their position as much as possible (see the derivatives course).

As a result of the limits to arbitrage, prices might not always reflect the fundamental value.

DeLong, Shleifer, Summers and Waldman (DSSW), 1990

DSSW tried to formalise the noise trader risk. From this point onwards, it becomes very technical. However, it would make a great impression if you get the general idea. They distinguish two types of investors:

  • Sophisticated investors, who have rational expectations, denoted by i.
  • Noise traders, denoted by n.

In the total market, quantified by the number 1, the noise traders are present in measure Quicklatex. Com e05fbada486d1c9c57340f498afd2bbc l3 and the sophisticated traders are present in measure Quicklatex. Com c1dc5a78e7bb505c6fc0e221090b6cc9 l3. As a starting point DSSW take the safe asset S that pays a fixed dividend r. The safe asset S is in a perfectly elastic supply, which means that a unit of it can be created out of a unit of consumption good in any period and a unit can be turned back into a unit of the consumption good in any period. The price of this theoretical safe asset is fixed at 1.

The other asset, U, is an unsafe asset and pays the same fixed real dividend r as asset S. The asset U is not in a perfectly elastic supply, which means that it’s in a fixed and unchangeable quantity, normalised at one unit. The price of U, in period t, is denoted as Quicklatex. Com 21e0f72624a047f5b65129487da5f4a2 l3.

If the price of each asset were equal to the net present value of its future dividend, then asset U and S would be perfect substituted and their prices would be the same in all periods. However, this is not how the price of U is determined in the presence of noise traders.

The sophisticated investor, in period t, accurately perceives the distribution of returns from holding the risky asset and therefore maximises the expected utility given the distribution. The noise trader, in period t, misperceives the expected price of the risky asset by an independent and identically distributed normal random variable Quicklatex. Com a8aa01c6d797037c4304cf673ed3dd44 l3.

(1)   Quicklatex. Com 30763cc9ec58e367d598d4a0df5d07d0 l3

The mean perception, Quicklatex. Com 749c1a3c5e4eb493ebdf72c92b50790c l3, is a measure of the average ‘bullishness’ of the noise trader and Quicklatex. Com d5c22f77a983f053d8caba47505036ba l3 is the variance of the noise traders’ misperception of the expected return per unit of the risky asset. Noise traders thus maximise their own expectation of utility given the next-period dividend, the one-period variance of Quicklatex. Com 2fd3f03da0d8b455b573c97ff6474c2f l3 and their false believe that the distribution of the price of U next period has mean Quicklatex. Com a8aa01c6d797037c4304cf673ed3dd44 l3 above its true value.

Each investors’ utility is a constant absolute risk aversion function of wealth when old, given by

(2)   Quicklatex. Com a8c99a75f38213c6a97ac8ffc75147d2 l3

In this equation, Quicklatex. Com ec88156094a620d1905abcc3fbc4b17d l3 is the coefficient of absolute risk aversion and W is the expected final wealth. Maximising U is equivalent to maximising Quicklatex. Com 2b4b3c40c5924cdec4d8f0d8b59ef652 l3, where Quicklatex. Com dcf699c1531e8d99033b927bc77fc290 l3 is the one period ahead variance of wealth.

  • The sophisticated investor chooses the amount Quicklatex. Com 3b5f2495b6de5e0554b6eb7b8c1e0f43 l3 of the risky asset U held to maximise

    (3)   Quicklatex. Com c8e92116f02b04462c4ce276b2838b15 l3

    (4)   Quicklatex. Com 9efc939336276bb9b40ffb3b6600ac3e l3

    Quicklatex. Com ecc2b3f17c485475900e166c879a9085 l3 is a function of first-period labor income.
  • The noise trader maximises Quicklatex. Com 847786156ea684fee207deb334e83924 l3 as

    (5)   Quicklatex. Com 6044df5dcf3c7c6faef9a6f554e21df0 l3

    The last term, Quicklatex. Com 414063496d87a19c2327f248bd365f57 l3, captures the noise traders’ misperception of the expected return from holding Quicklatex. Com ad36bbea22bbeb0a9834aee3275e2275 l3 units of the risky asset.

The young investors divide their portfolios between assets U and S. The quantities Quicklatex. Com 3b5f2495b6de5e0554b6eb7b8c1e0f43 l3 and Quicklatex. Com ad36bbea22bbeb0a9834aee3275e2275 l3 of the risky assets are functions of its price Quicklatex. Com 21e0f72624a047f5b65129487da5f4a2 l3 for the sophisticated trader and additional Quicklatex. Com a8aa01c6d797037c4304cf673ed3dd44 l3 for the noise traders. The optimal demand is a result from

(6)   Quicklatex. Com af0acaf7a2275da8df9effa1f762fc82 l3

  • Optimal demand for risky assets from the sophisticated investor is given by

    (7)   Quicklatex. Com 6473a84151c2b26be3ff519392641021 l3

  • Optimal demand for risky assets from the noise investor is given by

    (8)   Quicklatex. Com 650956326ed4064cf4d313d660038e3a l3

The theory continues by equating the demand from the young generation to the supply from the old generation.

(9)   Quicklatex. Com 8d33720534e40af0fcf4452ebb7fa011 l3

(10)   Quicklatex. Com 90770129bf0d6447c3544775c6ab2e3d l3

(11)   Quicklatex. Com daf59bdeb9bef76f3f051eff7ec0fa52 l3

This leads to the following equation regarding the price.

(12)   Quicklatex. Com 5ba46edfa6210f8058e1476788d296f3 l3

(13)   Quicklatex. Com 43dd2323342bf5010314d69123197b6a l3

The only uncertainty in the price is the noise traders’ misperception, where

(14)   Quicklatex. Com c517b74493927078fd1eb49bf5549b71 l3

Substituting Equation 14 into Equation 13 leads to

(15)   Quicklatex. Com c633f7e0a96ab78b78290d537c57a8b7 l3

Let’s discuss the terms in Equation 15.

  • The second term, Quicklatex. Com 3aaac3ff07699fd48f7158c987013751 l3, captures the fluctuation in the price of the asset U due to the variation of the noise traders’s misperceptions. When a generation of noise traders is more bullish than the average generation, they bid up the price.
  • The third therm, Quicklatex. Com e28c25f4aab9addedabe4261784cd190 l3, is called the price pressure effect. It captures the deviation of Quicklatex. Com 21e0f72624a047f5b65129487da5f4a2 l3 from its fundamental value due to the fact that the average misperception by noise traders is not zero.
  • The fourth term, Quicklatex. Com 457773ab8d56844d7082ba5a4d3b024a l3, is called the create space effect. It’s a compensation for the risk, where the investors believe that the asset is mis-priced.

Relative returns of noise traders and sophisticated traders

If noise traders’ portfolios are concentrated in assets that are subject to noise trader risk, noise traders can earn a higher average rate of return on their portfolios than sophisticated investors. Call the difference in returns between the two types of investors Quicklatex. Com 8f06074782ad7b2f43664e316d2c1277 l3,

(16)   Quicklatex. Com bb2cf9bd97e2e59cbf8ccccaa34899a8 l3

The first term, Quicklatex. Com 7bf74dc634cecc87ac03198ccc499b8d l3, indicates the difference between the noise traders’ and sophisticated investors’ demand. This can also be written as

(17)   Quicklatex. Com 28815407c1e0eb3e91ffd35238813126 l3

The second term, Quicklatex. Com a6fd1279b44d7adb19d06bb619bbcddb l3, indicates the expected value of the excess return on the risky asset U. This may also be written as

(18)   Quicklatex. Com fea2a9df0e86eb57c99a5d1be8f95e07 l3

This leads to

(19)   Quicklatex. Com a2e2fdbaa1c8836c82a3cb716b14b220 l3

Finally, as a result the given distribution in Equation 1,

(20)   Quicklatex. Com 4c1f92b62919fefba63a512e4fc86b96 l3

Let’s take a look at the terms in Equation 20.

  • The first term, Quicklatex. Com 749c1a3c5e4eb493ebdf72c92b50790c l3, increases the noise traders’ expected returns through what might be called the hold more effect.
  • Quicklatex. Com 862cc76441db82800087c033a24475aa l3 incorporates the price pressure effect. As noise traders become bullish, they demand more of the risky asset on average and drive up the price. As a result, they reduce the return to risk bearing which reduces the difference in returns between the sophisticated investors and noise traders.
  • Quicklatex. Com abd6836d88a5ff1b4f36a0c5d5220181 l3 incorporates the Friedmann effect, which is the risk for ‘buying high and selling low’. Because the noise traders; misperception is stochastic, they have the worst possible timing in the market. The later you step in, the more damage your poor market timing does with respect to the return.
  • Quicklatex. Com 566b409aad38d1b2011149172dc4751d l3 incorporates the create space effect. As the variability of noise traders’ believe increases, the price risk increases. To take advantage of noise traders’ misperception, stochastic traders must bear this greater risk. However, they are risk averse, so they reduce the extend to which they bet against noise traders. If the create space effect is large, then the price pressure and Friedmann effect inflict less damage on noise traders’ average returns relative to sophisticated investors’ returns.

From this lesson, we can draw the following conclusions.

  • The hold more- and create space effect tend to have a positive effect on Quicklatex. Com 8f06074782ad7b2f43664e316d2c1277 l3. The Friedmann- and price pressure effect have a negative effect. Neither of the pairs dominates the equation.
  • Depending on the average misperception, noise traders can have higher average returns than sophisticated traders.
  • Utility of sophisticated traders is by definition higher than the utility of noise traders. The demand for unsafe assets are derived from maximising the utility function. So, the misperception creates a bias.
  • The extra return does not compensate for the extra risk for noise traders, as it results in a lower utility.