**"assumptions"**

## 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!

## Capital Asset Pricing Model, Part Three – Other Assumptions

*by Eric Bank on January 22, 2011 · No Comments · in Documenting the Search for Alpha*

In Parts One and Two of our examination of the Capital Asset Pricing Model (CAPM), we evaluated two major assumptions:

1) Market returns are properly modeled by a normal distribution

2) Beta (systematic risk) is the sole source of priced risk for an asset or portfolio of asset

In Parts One and Two of our examination of the **Capital Asset Pricing Model (CAPM)**, we evaluated two major assumptions:

1) Market returns are properly modeled by a **normal distribution**

2) **Beta (systematic risk)** is the sole source of priced risk for an asset or portfolio of asset

As you recall, we found several weaknesses in both assumptions as they may apply to hedge funds. This time, we’ll examine the remaining assumptions underlying CAPM, and see if each is reasonable when applied to hedge fund trading.

CAPM assumes: Continue reading “Capital Asset Pricing Model, Part Three – Other Assumptions” »

## Capital Asset Pricing Model, Part Two – Systematic Risk

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

We are reviewing the underlying assumptions made by the Capital Asset Pricing Model (CAPM). Recall from last time the assumption that returns are distributed normally (i.e. a bell-shaped curve) and how this fails to account for skew and fat tails. Today we’ll look at CAPM’s assumption that there is but a single source of priced systematic risk: market beta.

We are reviewing the underlying assumptions made by the **Capital Asset Pricing Model (CAPM)**. Recall from last time the assumption that returns are distributed normally (i.e. a bell-shaped curve) and how this fails to account for **skew** and **fat tails**. Today we’ll look at CAPM’s assumption that there is but a single source of priced **systematic risk**: market **beta**.

Beta, of course, refers to the correlation between the returns on an asset and the returns on the broad market. Specifically, it is defined as:

βi = Cov(Rm, Ri) / Var(Rm) for asset i

that is, the **covariance** of the asset’s returns with the those of the market divided by the **variance** of the market returns. Another way to think about beta is as a discount factor: when divided into an asset’s expected return, it adjusts the expected return so that it equals the market’s expected return.

By making this assumption, CAPM is able to simplify the universe of different risks undertaken by a portfolio, such as a hedge fund, to one global number. That’s good for ease of calculation but bad for reflecting reality. It is also assumed that all non-systematic risk (**alpha**) has been diversified away by having a sufficiently large portfolio, so that the expected return on the portfolio is a function of its beta alone.

Many academics challenge the use of variance in the formula for beta. They point out that variance is a symmetrical measure of risk, whereas investors are only concerned about loss (just the left side of the curve). Instead of variance, they point to other measures of risk, such as **coherent risks**, that may better reflect investor preferences. Coherent risks have four attributes:

1) When comparing two portfolios, the one with better values should have lower risk

2) The joint risk of two portfolios cannot exceed the sum of their individual risks

3) Increasing the size of positions within a portfolio increases its risk

4) Picture two identical portfolios except one contains additional cash. The one with the extra cash is less risky, to the exact extent of the extra cash.

Hedge funds clearly take on risks that are not reflected in beta. These risks are the basis for the returns of many strategies, such as long/short (liquidity risk), fixed income (credit risk), and sector trading (sector risk) to name a few. To the extent that a hedge fund takes on risks that are not reflected by beta, it is modifying its long-term expected return for better or worse. For instance, a portfolio may add an asset with a negative expected return in order to hedge another asset that has an expected positive return. This has the effect of lowering the expected return of the portfolio as it lowers the portfolio’s risk.

Recall that the expected market return above the risk-free rate is called the market’s **risk premium**. Many hedge funds seek returns independent of the market risk premium. Rather, hedge funds supposedly choose their risk exposures, and hence their risk premia. Ideally, their market beta would be zero, so that hedge fund returns would reflect only risks other than market risk. Note that these non-market risks are often hard to diversify away. For instance, it is difficult to diversify away the risks involved in “risk arbitrage” – a misnomer for the strategy of taking positions on mergers and acquisitions. By assuming this “event risk” (i.e. by betting on whether a proposed merger will succeed), fund managers hope to capture the associated risk premium, independent of market beta.

There are a large number of hedge fund strategies, and most are exposed to their own systematic risks. Later on, we will review different strategies in detail, and look at the associated risk premia at that time. *The important point here is that systematic risk, whether captured in market beta or due to strategy-specific risk(s), should not be conflated with *alpha* – the returns due to superior skill and/or timing.* Ultimately we will be exploring the case for and against alpha when we examine **hedge fund replication strategies**.

Next time, we’ll finish discussing the assumptions underlying CAPM.

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