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Last time we began our discussion of Behavioral Portfolio Theory with a look at Safety-First Portfolio Theory (SFPT), which basically posits investor motivation to be to avoid ruin.  An extension to SFPT was introduced in 1987 by Lopes, named SP/A Theory. The letters in its title can be summarized:

S = security, a general concern about avoiding low levels of wealth

P = potential, reflects the general desire to maximize wealth

A = aspiration, the desire to reach a specific goal, such as achieving no less than the subsistence level s.

SFPT is a psychological theory of choice under uncertainty. In Lopes’ framework, risk-taking is balanced between fear and hope. Lopes posits that fear is such a strong factor because fearful people overweight the probability of the worst outcomes, underweight those for the best outcomes. This leads individuals to understate the probability of achieving the highest level of expected wealth E(W).  In other words, fearful individuals are pessimistic.

Hope has the inverse effect on individuals – optimism causes hopeful investors to overstate the probability of achieving the highest level of expected wealth.

For those of you who like symbols, wealth W is actually a series of n different levels of wealth, ordered from lowest to highest, designated as W1, W2, W3, … Wn-1, Wn. Lopes denotes a decumulative distribution function[1] D(x) as the likelihood that the wealth W of a portfolio will be greater than the investor’s subsistence level s:

D(x) = Prob {W >= s)

where 0 < x <= n different levels of wealth.

Expected wealth E(W) is the weighted sum of each level of wealth, the weight being the probability pi of achieving a particular level of wealth Wi:

E(W) = ∑piWi

Lopes notes that expected wealth, E(W) = ∑piWi, can be expressed as the sum of the decumulative probabilities ∑Di (Wi - Wi -1), where the summation is from i = 1 to n and W0 is zero. In this expression for E(W),the individual receives W1 with certainty (note that decumulative probability D1 = 1), receives the increment W2 – W1 (that is, an amount over W1) with probability D2, receives the further increment W3 – W2 with probability D3, and so on. Lopes contends that fear operates through an overweighting of the probabilities attached to the worst outcomes (W1). The way that works is that fearful investors set the p1 too high, and pn too low. The reverse is true for hopeful investors.

Lopes concludes that the emotions of fear and hope reside within all individuals, and that each emotion serves to modify the decumulative weighting function D. She suggests that the final shape of the decumulative transformation function h is a convex combination (shaped like a smile) of hs (for fearful investors worried about security) and hp (for hopeful investors looking for maximum potential), reflecting the relative strength of each.

The qualitative lesson from SP/A theory is that you should establish a portfolio in which you have reigned in your emotions of fear and hope, so that neither dominates. Unfortunately, it doesn’t tell you how to accomplish this. Still, not a bad lesson for a hedge fund trader.


[1] The decumulative distribution function is just (1 – cumulative distribution function).  Basically it measures the curve to the right (instead of the left) of a given point.

Non-diversifiable risk

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.

Prime brokers offer a variety of services to investors, from providing credit to clearing trades. One important service offered is known as Securities Lending. In Part One of this article, we’ll look at the contractual and collateral rules pertaining to Securities Lending.

As an investor or hedge fund, you may wish to borrow shares for a variety of reasons, such as shorting the stock or hedging a long position. An executed Securities Lending Agreement is the documentation required before shares are loaned. Continue reading “Securties Lending, Part One” »

Picture of the U.S. Treasury Building in Washington D.C.

U.S. Treasury

Repurchase agreements are contracts involving the simultaneous sale and future repurchase of an asset, most often Treasury securities.  Typically, the seller buys back the asset at the same price at which it sold. On the buy-back date, the original seller pays the original buyer interest on the implicit loan created by the transaction.  Interest due on a repo at maturity is at the rate for the stated maturity of the repo.

A reverse repo, or simple a reverse, is a contract in which a repo is structured so that a broker/dealer, a bank, or another party that normally uses the repo market to fund (finance) itself is cast in the role of securities purchaser and money lender  Broker/dealers often cover shorts by reversing in securities.

A sell/buyback is essentially the same as a repo, except for the treatment of coupon interest. Coupon payments are not forwarded to the investor by the counterparty. Instead, the counterparty pays the investor repo interest on the coupon payment. When the sell/buyback terminates, the investor will receive its accrued coupon interest from the counterparty. A sell/buy back transaction also differs from a repo transaction in that the sales price for securities delivered differs from the purchase price paid when the securities are returned.  The difference in the price of the sale and purchase transactions accounts for the amount of accrued coupon interest earned and the financing cost charged during the term of the loan.

A buy/sellback is similar to a reverse repo. Continue reading “Repurchase Agreements” »

In our previous blog, we discussed the concept of beta as it applies to the risk and return of an investment. Recall that beta is the price movement in an individual investment that can be accounted for by the price movement of the general market.  If your investment has a beta of 1.0 and the market returns 10%, your investment should also return 10%.  If your investment returns over 10%, the excess return is called alpha. Alpha is derived from a in the formula Ri = a + bRm which measures the return on a security (Ri) for a given return on the market (Rm) where b is beta.

Continue reading “Risk and Return: Alpha” »