Arbitrage is a widely used trading strategy and probably one of the oldest trading strategies available. The concept is closely linked to the market efficiency theory. The theory says that before the market is really effective, there must be no arbitrage options – all corresponding assets must go the same way in terms of price.

Although a simple strategy, very few – if any – investment funds rely solely on such a strategy. That fact can be explained by the difficulties of taking advantage of the usually short-lived situation. With the rise of electronic commerce, which can execute trading orders within a fraction of a second, incorrectly set price differences occur in a smaller period of time. The improved trading speed has thus improved the possibility of implementing the arbitrage strategy.

Arbitrage is generally exploited by large financial institutions because it requires significant resources to identify the options and execute the trades. They are often executed using complex financial instruments, such as derivative contracts and other forms of synthetic instruments, to find similar assets. Derivative trading often involves margin trading and a large amount of cash required to execute the trades.

Paintings are alternative assets with a subjective value and tend to give rise to arbitrage opportunities. For example, a painting’s paintings can be sold cheaply in one country, but in another culture, where their painting is more valued, sold for significantly more. An art dealer can thus arbitrate, by buying the paintings where they are cheaper, and selling them in the country where they bring a higher price.

As day trader/momentum trade we work fast, looking to make lots of little profits by trading stocks and other securities during a single day. *Arbitrage* is a trading strategy that looks to make profits from small discrepancies in securities prices. We trade stocks in US and Denmark.

The idea is that the *arbitrageur,* or *arb* (the person who does arbitrage/trading), arbitrates among the prices in the market to reach one final level. In theory, arbitrage is riskless. It’s illogical for the same asset to trade at different prices, so eventually the two prices must converge. The person who buys at the lower price and sells at the higher one makes money with no risk. Arbitrage can also be used to move towards the market and expose itself in a short position where one bets on price declines. By buying expensive and selling cheaper and taking a small difference home.

In academic theory, markets are perfectly efficient, and arbitrage simply isn’t possible. That makes a lot of sense if you are testing different assumptions about how the markets would work in a perfect world.

A long-term investor would say that markets are inefficient in the short run but perfectly efficient in the long run, so they believe that if they do their research now, the rest of the world will eventually come around, allowing them to make good money.

Traders are in between. The market price and volume are pretty much all the information they have to go on. It may be irrational, but that doesn’t matter today. The only thing a trader wants to know is whether an opportunity exists to make money given what’s going on right now.

In the academic world, market efficiency comes in three flavors, with no form allowing for arbitrage:

Arbitrage is a widely used trading strategy and probably one of the oldest trading strategies available.

The concept is closely linked to the market efficiency theory. The theory says that before the market is really efficient, there must be no arbitrage possibilities – all corresponding assets must go the same way in terms of price, ie. it must be in an upward trend. We are trading in an upwarding trend to make sure that the bottom is increasing. The similarity of prices in different markets measures market efficiency.

The Capital Asset Pricing Model (CAPM):

In the 1960s, William Sharpe and John Lintner introduced the world of CAPM after working on Harry Markowitz’s mean-variance optimization portfolio theory. The purpose of CAPM is to describe the relationship between the systematic market risk and the expected return on an asset when the risk-free interest rate is taken into account.

Understanding the Capital Asset Pricing Model (CAPM) The formula for calculating the expected return of an asset given its risk is as follows:

–

*ERi**=**Rf**+**βi**(**ERm**−**Rf**)*

*where:*

*ERi **= **expected return of investment*

*Rf* *= **risk-free rate*

*β**i **= **beta of the investment*

*(**ERm**−**Rf**) **= **market risk premium.*

–

Investors expect to be compensated for risk and the time value of money. The risk-free rate in the CAPM formula accounts for the time value of money. The other components of the CAPM formula account for the investor taking on additional risk. The beta of a potential investment is a measure of how much risk the investment will add to a portfolio that looks like the market. If a stock is riskier than the market, it will have a beta greater than one.

If a stock has a beta of less than one, the formula assumes it will reduce the risk of a portfolio. A stock’s beta is then multiplied by the market risk premium. (The market risk premium is the difference between the expected return on a market portfolio and the risk-free rate. The market risk premium is equal to the slope of the security market line (SML), a graphical representation of the capital asset pricing model (CAPM). CAPM measures the required rate of return on equity investments, and it is an important element of modern portfolio theory and discounted cash flow valuation.). The risk-free rate is then added to the product of the stock’s beta and the market risk premium. The result should give an investor the required return or discount rate they can use to find the value of an asset. The goal of the CAPM formula is to evaluate whether a stock is fairly valued when its risk and the time value of money are compared to its expected return. For example, imagine an investor is contemplating a stock worth $100 per share today that pays a 3% annual dividend. The stock has a beta compared to the market of 1.3, which means it is riskier than a market portfolio. Also, assume that the risk-free rate is 3% and this investor expects the market to rise in value by 8% per year. The expected return of the stock based on the CAPM formula is 9.5%:

*9.5%=3%+1.3×(8%−3%).*

Where E (Rp) is the expected return on the portfolio, rf is the risk-free interest rate and β [E (rm) – rf] is a market risk premium which must be a compensation for holding risky assets. The model quickly became popular as it is relatively simple and this means that it is still used in recent times. There are some assumptions behind the model: all investors choose a mean-variance efficient portfolio and have an investment horizon of one period, all investors have the same perception and understanding of information and thus analyze assets in the same way and a fully efficient market without transaction costs and restrictions on credit facilities.

These assumptions do not apply in the real world. Fama and French (2004) used the CAPM to do an empirical analysis and concluded that it performs poorly, which may be due to the assumptions. Fama-French 3-Factor Model Due to the poor empirical results, economists would find other factors that could systematically explain the return on financial assets. In 1992, two factors were documented, the size and value of the company that could help explain the return. Eugene Fama and Kenneth French (1992; 1993) found that these two factors could supplement the market risk of explaining stock returns. The model looks like this:

–

*R*it−*Rft*=*αit*+*β*1(*R*Mt−*Rft*)+*β*2SMBt+*β*3HMLt+*ϵit*

where:

*R**it *= total return of a stock or portfolio *i* at time *t*

*R**ft *$= risk free rate of return at time t$

*R**Mt *$= total market portfolio return at time t$

*R**it *$−Rf t= expected excess return$

*R**Mt *$−Rft =excess return on the market portfolio (index)$

*SMB**t *$= size premium (small minus big)$

*HML**t *$= value premium (high minus low)$

$β1,2,3 = factor coefficients$

Where Ri – rf is an excess return on portfolio, αi is the intersection of the Y axis and β [E (rm) – rf] is the market risk premium from CAPM and ei is faulty. The two new factors are Small Minus Big (SMB) and High Minus Low (HML) factor premiums. SMB is a size factor which is constructed based on the size of the companies. HML is a value factor that is constructed on the basis of the companies’ carrying value in relation to market value.

The Carhart 4-factor model In 1993, the first trends of a momentum factor were analyzed. Narasimhan Jagadeesh and Sheridan Titman (1993) analyzed that an investment strategy consisting of buying shares that had performed relatively well over the past 3-12 months and selling shares that had performed relatively poorly over the past 3-12 months would create positive stock returns. Mark Carhart used Mark Carhart this finding to create a new factor for the Fama-French model, a momentum factor. This will make the model look like the following:

**Required trading conditions:**

Arbitrage can occur if the following conditions are met:

Asset Price Imbalance: This is the primary condition of arbitrage. Price imbalance can take various forms:

– In different markets, the same asset is traded at different prices.

– Assets with similar cash flows are traded at different prices.

– An asset with a known future price is currently traded at a price that differs from the expected value of future cash flows.

Simultaneous trading execution: Purchase and sale of identical or equivalent assets should be performed simultaneously to capture the price differences. If the transactions are not executed simultaneously, the trade will be exposed to significant risks.

While this is a simple strategy, very few – if any – investment funds rely solely on such a strategy. This fact can be explained by the difficulties of exploiting the generally short-lived situation. With the rise of electronic commerce, which can execute trading orders within a fraction of a second, incorrect price differences occur in a smaller period of time. The improved trade speed has in this sense improved the efficiency of the markets.

In addition, equal assets with different prices generally show a small price difference, less than the transaction costs of an arbitrage trade would be. This effectively eliminates the arbitrage option.

Arbitrage is generally exploited by large financial institutions because it requires significant resources to identify the opportunities and execute the trades. They are often executed using complex financial instruments, such as derivative contracts and other forms of synthetic instruments, to find similar assets. Derivative trading often involves margin trading and a large amount of cash required to execute the trades.

Paintings are alternative assets with a subjective value and tend to give rise to arbitrage opportunities. For example, a painter’s paintings can sell cheaply in one country, but in another culture, where their painting is more valued, sell for significantly more. An art dealer could arbitrate by buying the paintings where they are cheaper and selling them in the country where they bring a higher price.

One of the first things to understand about arbitrage pricing theory is that the concept has to do with the process of asset pricing. Essentially, the arbitrage price theory, or APT for short, helps establish the pricing model for various stocks. Here is some information on the arbitrage pricing theory and why this concept is so useful in determining the pricing model for buying and selling stocks.

Developed by Economist Stephen Ross in 1976, the underlying principle of pricing theory involves the recognition that the expected return on any asset can be mapped as a linear calculation of relevant macroeconomic factors related to market indices. It is expected that there will be some rate of change in most, if not all, of the relevant factors. Running scenarios using this model helps reach a price that matches the expected outcome of the asset. The desired result is that the asset price will match the expected price at the end of the quoted period, with the final price discounted at the rate implied by the Capital Asset Pricing Model. If the asset price is off course, arbitrage will help bring the price back into reasonable perimeters.

In practical application, the use of the arbitrage pricing theory can work very well in increasing the long-term value of a stock portfolio. For example. The use of APT when the current price is very low will result in a simple process that would yield a return but still keep the portfolio safe. The first step would be to short sell the portfolio and then buy the low-priced asset with the proceeds. At the end of the period, the low price, which will have increased in value, will be sold and the proceeds used to buy the recently sold portfolio. This strategy usually results in a modest amount of income for the investor.

A similar strategy is used when the current price is high. Under this set of circumstances, the investor would briefly sell the expensive asset and use the proceeds to buy the portfolio. At the end of the period, the investor then sold the portfolio, used a portion of the proceeds to buy back the high price asset, and again made a profit for the transaction.

The use of arbitrage pricing theory is very common in the stock market today. In some ways, the use of the theory is even more widespread than ever, as more investors have access to real-time information via online methods than at an earlier point in trading history. As a means of analyzing current conditions and reacting accordingly, the arbitrage pricing theory has a solid result of the results and will no doubt be employed by investors and analysts for many years to come.

**Assumptions in arbitrage pricing theory**

Arbitrage pricing theory works with a pricing model that factors in many sources of risk and uncertainty. Which we at Pillious is trying to learn from the theory, to take more factors into consideration before investing. Unlike the Capital Asset Pricing Model (CAPM), which only considers the single risk level factor in the overall market, the APT model looks at several macroeconomic factors that, in theory, determine the risk and return on the specific asset.

These factors provide risk premiums for investors to consider because the factors carry systematic risk that cannot be eliminated by diversifying.

Arbitrage pricing theory and Pillious suggests that investors will diversify their portfolios, but that they will also choose their own individual profile of risk and return based on premiums and sensitivity to the macroeconomic risk factors.

**A more complex example**

A very common example used to describe arbitrage opportunities is with cross-border listed companies. Let’s say one person owns shares of Company ABC, listed on Canada’s TSX, trading at $ 10.00 CAD. At the same time, the ABC stock is listed in NYSE trades at $ 8.00 USD. The current CAD / USD exchange rate is 1.10. One trader could buy shares of NYSE for $ 8.00 USD and sell shares of TSX for $ 10.00 CAD. This would give him a profit of $ 1.09 USD per Share.

Mathematical model for APT:

Arbitrage pricing theory can be expressed as a mathematical model:

ER (x) – Expected return on the asset

Rf – Risk-free return

β n (Beta) – The price sensitivity of the asset to factor

RP n – The risk premium associated with factor

Historical return on securities is analyzed with linear regression analysis against the macroeconomic factor to estimate beta coefficients for the arbitrage pricing theory formula.

**Entries in the Arbitrage Pricing Theory**

Arbitrage pricing theory provides more flexibility than CAPM; however, the former is more complex. Inputs that complicate the arbitrage pricing model are the asset’s price sensitivity to factor n (β n) and the risk premium for factor n (RP n).

Before the investor chooses a beta and risk premium, the investor must choose the factors that they believe influence the return on the asset; it can be done through basic analyzes and a multivariate regression. One method of calculating the beta for the factor is to analyze how the beta affected many similar assets / index and get an estimate by running a regression on how the factor affected the similar assets / index.

A lot of the factors and theories are well documented and with a lot of time to analyze, it can be put into a trading strategy that goes beyond Arbitrage strategy. Instead a strategy that is possible to follow along with, and make a good solid return on. That strategy is what we use at Pillious, where you have the possibillity to follow along. Where we will analyze and factor many things, to obtain a greater advantage by lowering the potential risk.

In the Arbitrage strategy, the risk premium can be obtained by comparing the historical annual return of the similar assets / indices with the risk-free rate, adding betas for the factors multiplied by the factor premiums and solving for the factor premiums.

**Example**

Suppose that:

You want to apply the arbitrage pricing theory to a well-diversified portfolio of stocks.

The safe return is 2%.

Two similar assets / indices are the S&P 500 and the Dow Jones Industrial Average (DJIA).

Two factors are inflation and gross domestic product (GDP).

Betas of inflation and GDP of S&P 500 are 0.5 and 3.3 * respectively.

Betas of inflation and GDP at DJIA are 1 and 4.5 * respectively.

The S&P 500 expected return is 10% and DJIA’s expected return is 8% *.

* Betas do not represent actual betas in the markets. They are used for demonstrative purposes only.

* Expected return does not represent the actual expected return. They are used for demonstrative purposes only.

**After settling for the risk premiums, we are left with the following for our well-diversified portfolio: (Right side)**

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