**"arbitrage pricing theory"**

## Behavioral Portfolio Theory 1 – Safety First

*by Eric Bank on February 18, 2011 · No Comments · in Documenting the Search for Alpha*

ur survey of portfolio theories continues; we have already evaluated Modern Portfolio Theory, the Capital Asset Pricing Model and the Arbitrage Pricing Theory in earlier blogs, and now turn to a series of articles on Behavioral Portfolio Theory (BPT).

Our survey of portfolio theories continues; we have already evaluated **Modern Portfolio Theory**, the **Capital Asset Pricing Model** and the **Arbitrage Pricing Theory** in earlier blogs, and now turn to a series of articles on **Behavioral Portfolio Theory (BPT)**. Recall that as we continue to lay out a financial framework, our ultimate goal is to figure out whether hedge funds actually provide alpha returns to investors.

Whereas both CAPM and ABT assume that investors are proceeding from a *single unified motivation* – to maximize the values of their portfolios – BPT instead suggests that investors have a *variety of motivations* and structure their investments accordingly. Think of a layered pyramid, in which the base is set up to protect against losses and the top designed for high reward. This has been characterized as the “government bond and lottery ticket” portfolio because of the extreme divergence of goals.

One of the earliest attempts at BPT was **Safety-First Portfolio Theory (SFPT**, Roy 1952). The theory posits that investors want to minimize the probability of **ruin**, which is defined as when an investor’s **wealth W** falls below his **subsistence level s**:

**min Pr(W < s)**.

SFPT, like the other portfolio theories, makes a few assumptions:

- a portfolio has a return
**mean****μ**and return_{P}**standard deviation****σ**;_{P} - the
**risk-free asset R**is not available;_{F} - investors trade off assets according to the
**expected utility**of the assets and the law of diminishing returns; - the subsistence level s is
**low**; and - the distribution of portfolio returns is
**normal**(bell-shaped).

In short, an investor wants to own a portfolio for which

**(s – ****μ _{P}) / σ_{P}**

_{ }

is minimized.

Later refinements of the theory included:

- relaxing the fixed value s;
- constraining the probability of ruin such that it never exceeds a predetermined ruin probability
- expecting investors to maximize wealth within the safety-first constraint

As an early behavioral portfolio theory, SFPT has a somewhat simplified vision of investor as primarily risk-adverse. Also, it postulates that investors follow utility theory in their decision-making, an assumption widely discredited by empirical studies.

We progress next time to a theory that had its roots in SFPT: **SP/A Theory,** which deals with security, potential, and aspiration. Don’t miss it!

## Arbitrage Pricing Theory – Part Three

*by Eric Bank on February 14, 2011 · No Comments · in Documenting the Search for Alpha*

Having previously discussed them, let’s compare them. How does the Arbitrage Pricing Theory (APT) stack up against the Capital Asset Pricing Model (CAPM)?

Having previously discussed them, let’s compare them. How does the **Arbitrage Pricing Theory (APT) **stack up against the **Capital Asset Pricing Model (CAPM)**?

Both theories are important mechanisms for pricing assets. The APT assumes less than CAPM. Whereas CAPM relies on probabilities and statistics, APT provides for cause and effect to explain its predictions of asset returns. CAPM assumes investors hold a mix of cash and the market portfolio, whereas APT allows each investor to hold his own portfolio with its own unique beta. So if you applied APT to the market portfolio, the example would degenerate into CAPM, because the securities market line would be a single factor price model in which beta is exposure to price changes in the market portfolio.

Since the beta in CAPM is correlated to asset demand based on each investors incremental utility for each asset, it is considered a demand-side model. In contrast, APT is a supply-side model, because its betas correlate an asset’s return to specific economic factors. Any sudden change to one of these factors would ripple through the market containing the asset, and thus are inputs to asset returns.

Both models assume **perfect competition** within markets:

- Buyers are
*price-takers*, and are too small to affect prices. - Assets are i
*nfinitely liquid*, and supply and demand will reach equilibrium at a certain price. - There are no
*barriers*to entering or leaving a market. - There are no
*trading costs*such as commissions, taxes or fees. - All investors share a sole motivation: to
*maximize returns*. - A given share of a security is the
*same*as any other share, regardless of where (i.e. on which exchange) the security is exchanged.

The only factor in CAPM is the market portfolio. ABT is a multi-factor model and has one additional requirement: the number of factors must be no greater than the number of assets. Certain assumptions are made about APT factors:

- The factors are
*macroeconomic*in nature, and thus create risks that cannot be easily avoided through diversification. - Factors affect asset prices through
*shocks*(unexpected changes). - All information about each factor is
*well-known and accurate.* - There is some real-world
*relationship*between a factor and an asset.

So, which model better predicts asset returns? The answer is not clear. Quoting from a study published in the Asia-Pacific Business Review, March, 2008 by Rohini Singh: “*The macroeconomic factors used in this study were able to explain returns marginally better [using APT] than [CAPM] beta alone. While this confirms that risk is multidimensional and that we should not depend on beta alone, further research is required to identify other variables that can help explain the cross section of returns.*”

## Arbitrage Pricing Theory – Part Two

*by Eric Bank on February 10, 2011 · No Comments · in Documenting the Search for Alpha*

Arbitrage Pricing Theory (APT) is a multi-factor model in which a series of input variables, such as macroeconomic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset.

In our previous blog on this subject, we identified **Arbitrage Pricing Theory (APT)** as a* multi-factor model* in which a series of input variables, such as macroeconomic indicators and market indices, are each assigned their own *betas* to determine the expected return of a target asset. Recall from its equation that the expected return of an asset *i *is a linear function of the assets’ betas to the *n* factors:

E(r_{i}) = R_{f} +b_{i1} RP_{1}+. . .+b_{in} RP_{n} where

- RP
_{n}is the premium on factor n - b
_{in}is asset i’s loading of factor n - R
_{f}is the risk-free rate

Let’s explore the theory in a little more depth. It is based on the concept of arbitrage, which is when a trader makes a risk-free profit when a single asset is priced differently in different markets. By shorting the higher price while simultaneously buying at the lower price, the price differential is locked in. Unfortunately, in today’s lightning-fast global markets, this is a very rare opportunity.

The theoretical value of the APT is its ability to detect a *mispriced asset*: if the weighted sum of each factor’s risk premium is different from the asset’s current risk premium (as indicated by the asset price), then an arbitrage opportunity exists. If you think as the expected return on an asset as a *discount rate* on its future cash flows, a price can be calculated and compared to the actual current price.

We use the term asset here to encompass both single assets and portfolios. If a single asset is being modeled, then the asset is exposed to each macroeconomic factor to the extent of its beta with that factor. Let’s assume we identify a mispriced asset based on a 5-factor ABT model. An *equivalent portfolio* would contain a set of 6 correctly-priced assets, one asset per factor plus one extra. Each asset would have the same exposure to a single factor as the mispriced asset. By varying the amount of each asset in the portfolio, you can arrive at a portfolio beta/factor that is the same as that for the mispriced asset. You can now treat the portfolio as a *synthetic* version of the asset, and take a short or long position in the portfolio to offset the opposite position in the mispriced asset, theoretically capturing a risk-free profit.

Of course, identifying the important factors associated with a particular asset is no small undertaking, and APT does not offer a roadmap. However in general, one would have to take into account, for each of the factors affecting a particular asset, the expected return of each factor and the asset’s sensitivity to the factor.

We’ll complete our look at APT next time by comparing it to the Capital Asset Pricing Model.

## Arbitrage Pricing Theory – Part One

*by Eric Bank on February 5, 2011 · No Comments · in Documenting the Search for Alpha*

Arbitrage Pricing Theory (APT) is a multi-factor model conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor.

Last time out, I promised we would look at the risk premia associated with various hedge fund strategies. But first I’d like to finish the discussion of pricing models that we began with our look at the **Capital Asset Pricing Model**.

**Arbitrage Pricing Theory (APT)** is a* multi-factor model *conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own *betas* to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor. Here are the equations:

Suppose that asset returns are driven by a few (*i*) common systematic factors plus non-systematic noise:

r_{i}= E(r_{i}) +b_{i}_{1} F_{1} + b_{i}_{2} F_{2} +· · ·+b_{in} F_{n} + ε_{i}_{ } for i = 1, 2, . . .n where:

- E(r
_{i})*i* - F
_{1}, . . ., F_{n}are the latest data on common systematic factors driving all asset returns - b
_{in}gives how sensitive the return on asset i with respect to news on the n-th factor (*factor loading*) - ε
_{i}is the idiosyncratic noise component in asset i’s return that is unrelated to other asset returns; it has a mean of zero

APT claims that for an arbitrary asset, its expected return depends only on its *factor exposure*:

If we define RP_{i} = the *risk premium *on factor i = the return from factor i above the risk-free rate = (rF_{n}−R_{f}), then the **APT equation** is:

E(r_{i}) = R_{f} +b_{i1} RP_{1}+. . .+b_{in} RP_{n} where

- RP
_{n}is the premium on factor n - b
_{in}is asset i’s loading of factor n - R
_{f}is the risk-free rate

So you can see, the expected return of an asset *i *is a linear function of the assets’ betas to the *n* factors. This equation assumes ideal markets and no surplus of the number of factors above the number of assets.

In Part Two, we’ll explore the assumptions and implications of the APT equation.

## Modern Portfolio Theory – Part One

*by Eric Bank on December 30, 2010 · No Comments · in Documenting the Search for Alpha*

Our risk/return series continues with a review of Modern Portfolio Theory (MPT). We’ve already looked at alpha, beta, efficient markets, and returns. Our ultimate goal is to evaluate the role of alpha in hedge fund profitability, how to replicate hedge fund results without needing alpha, and finally how you can start your own cutting-edge hedge fund using beta-only replication techniques.

Our risk/return series continues with a review of **Modern Portfolio Theory (MPT)**. We’ve already looked at **alpha, beta, efficient markets**, and **returns**. Our ultimate goal is to evaluate the role of alpha in hedge fund profitability, how to replicate hedge fund results without needing alpha, and finally how you can start your own cutting-edge hedge fund using **beta-only replication techniques**.

MPT suggests that a portfolio can be optimized in terms of risk and return by carefully mixing individual investments that have widely differing betas. Recall that beta is return due to correlation with an overall market (known as **systematic risk**). To take a trivial example, if you hold equal long and short positions in the S&P 500 index, your portfolio would have an overall beta of (.5 * 1 + .5 * -1) = 0. There would be no risk, but in this case, there would be no return either, except for the slow drain of commissions, fees, etc.

### Visit Eric on Google+

### Archives