Alpha’s residual nature

By Guy Lalonde | October 14, 2011 | Last updated on October 14, 2011
5 min read

This is part 2 in a series on alpha and beta.

In my last article, “The true cost of beta,” (June 2011), I noted that a portfolio manager’s return is made of two components: beta and alpha. Beta is the return you get by investing in various risk factors, like market risk.

Alpha is the extra return generated through manager skill above the return accounted for by the risk premium of the underlying factors the portfolio is exposed to. Beta is abundant and cheap, while alpha is a rare and fleeting commodity that fetches a higher price if it’s sustainable in the long run. Alpha is what is left after we’ve accounted for the beta portion of a manager’s return— it’s a residual value.

Alpha throughout history

Alpha is the much sought-after ability for a manager to generate excess return: the return above a benchmark portfolio. Indexes have been published from the early days of market trading, either by the exchanges themselves or by publishing companies such as Dow Jones and Standard & Poor’s, so these references became the benchmarks to beat. Thus one popular notion of alpha is that of a manager beating the S&P 500, or the S&P/TSX Composite Index, for example.

A theoretical reference portfolio first appeared in financial literature in the early 1960s: the Capital Asset Pricing Model, or CAPM. In a nutshell, the CAPM tells us that in the context of an efficient market, stock picking doesn’t work. If you want more return, you need to take on more risk (in the form of leverage). Any risk-adjusted return above this is called “abnormal return,” or alpha. In CAPM terms, return is tied to the notion of exposure to the market in general and represented by the linear regression model.

CAPM: E(Rp) = Rf + B x (Rm-Rf)


Rp: The portfolio return

E(Rp): The expected portfolio return as per CAPM

Rf: the risk-free rate

B: the manager’s Beta, or leverage factor

(Rm-Rf): the risk premium expected from investing the market.

CAPM is a single-factor equilibrium model for which the risk factor is market risk. In efficient markets, a manager’s return should be 100% dictated by and limited to exposure to the market risk factor. In other words, Rp = E(Rp).

In an efficient market—a market whose general conditions do not allow for added return through manager skill—a manager’s return is solely dependent on the extent of his beta (given a constant risk-free rate and a constant market risk premium), or exposure to the underlying risk factor.

So if a manager’s return in the last twelve months came in at 1% above his benchmark, the efficient market hypothesis tells you that’s because he took on more risk. If expected market return is 7% and the risk-free rate is 2%:

E(Rp) = 2% + 1 x (7%-2%) = 7%

If the manager’s return over twelve months is actually 8%, and his beta is 1.2 (he is leveraged 1.2 to 1):

E(Rp) = 2% + 1.2 x (7%-2%) = 8.0%

In this case the manager has not generated any skill-based return, as the actual portfolio return is equal to the expected portfolio return given a beta of 1.2:

Rp– E(Rp) = 8.0 – (2% + 1.2 x (7%-2%)) = 8.0 – 8.0 = 0

In other words, apart from the risk free rate, the manager’s return is completely explained by the degree of exposure to the market risk factor. There is no leftover return that needs explaining. If there were, under single-factor CAPM, we call that residual return alpha and attribute it to manager skill.

[α]p = Rp– E(Rp)


[α]p = Jensen’s alpha

Rp = Funds’ realized return

E(Rp) = Funds’ expected return

What if the manager’s beta is 1? In this case, CAPM is not able to explain all of the manager’s performance. There is a residual return that remains unexplained after taking into account the manager’s CAPM beta exposure and risk-free rate. That variable would be the ever-sought-after excess return, alpha, αp:

E(Rp) = Rf + [α]p + B x (Rm-Rf)

Our example above would look more like this:

8.0 – (2% + 1.0 x (7%-2%))

8.0 -7.0 = 1

[α]p = Rp– E(Rp)

1 = 8%-7% = [α]p

Notice that the manager has produced a return of 1% above the return expected under CAPM, and has done so without taking on additional beta.

But what if there are other risk factors to be considered? What if in addition to being exposed to the market risk factor, the manager is in fact specialized in small-cap stocks, which are riskier than large caps? Then the expected portfolio return, E(Rp), would look something like:

E(Rp) = Rf + B1 x (Rm-Rf) + B2 x (RS– RB)

So now we have two betas. The general market beta, and an additional beta for the additional risk factor taken on by investing in small-cap stocks versus safer large-cap stocks (Small minus Big, or SMB). Let’s assume the risk premium of owning small-cap stocks versus large-cap stocks is 1% per year. Plugging in the manager’s numbers would look like:

E(Rp) = 2% + 1 x (7%-2%) + 1 x (8%-7%)

E(Rp) = 8%

We see that expanding the value of beta actually reduces what was considered alpha under single-factor CAPM to zero.

Beta has evolved through time as new risk-factor models have been identified. Eugene Fama and Kenneth French expanded the original single-factor model to a three-factor model that includes a size factor (RS – RB) as well as a book-to-market factor (RH– RL, or high book value minus low book value):

E(Rp) = Rf + B1 x (Rm-Rf) + B2 x (RS – RB) + B3 x (RH– RL)

Mark Carhart later demonstrated the Fama and French model did not capture return attributable to the one-year return momentum of a stock. The Carhart four-factor model takes the Fama and French three-factor model and adds a momentum risk factor (Winners minus Losers, or RW– RL):

E(Rp) = Rf + B1 x (Rm-Rf) + B2 x (RS– RB) + B3 x (RH– RL)

Proliferation of ETFs

Since alpha is a residual return once all sources of beta are accounted for, any expansion of beta will make alpha more rare.

The slicing and dicing of various market betas such as size, value, sector concentration, style, geography, momentum and asset class, combined with the arrival to market of ETFs tracking these various risk factors, are steadily pushing alpha-chasing and accompanying fees to the sidelines, or at least giving them a complementary role in portfolio building as opposed to the central objective.

The investor and advisor can build high-performing and low-cost portfolios consisting of diversified factor exposure and go from there. Taking each of the above factor models, we can find at least one corresponding low-cost ETF offering exposure to the appropriate risk factor:

Market risk premium: Vanguard S&P 500 ETF: VOO (0.06%)

Small minus big: iShares Russell 2000 ETF: IWM (0.26%)

Three-factor model: iShares Russell 2000 value ETF: IWO (0.40%)

Strong momentum stocks:

PowerShare DWA technical leaders portfolio ETF: PDP (0.70%)

In these cases, what was once considered alpha and priced accordingly is now considered unrecognized beta, and priced at a much lower level.

As mentioned in my last article, beta occurs on a continuum from pure or classical beta to more complex risk-factor exposure. The cost of more complex beta, unsurprisingly, is higher than the cost of classical beta.

But while there’s nothing wrong with paying extra for genuine alpha-generation effort, there is no reason for the investor to pay alpha-like fees for a product that delivers beta, classical or not.

Guy Lalonde, is an investment advisor and portfolio manager at National Bank Financial and is based in Pointe Claire, Quebec.

Guy Lalonde