# Definition of factor premium: against cap weighted index or against treasury bills?

How can we define a factor premium? In the book Berkin & Swedroe: Your Complete Guide to Factor-Based Investing the authors start introducing the market bet premium and define it as the average return of the total US stock market minus the average return of Us treasury bills overa certain period.

For other factors (size, value, momentum) if I read it correctly they use the same definition: average factor return minus average return of US treasury bills.

In the case of the market the definition makes sense: how much do I earn if I invest in stocks instead of the risk less asset (edit thanks to Chris Taylor).

Factors in their setting are defined in the long/short version (e.g. small minus big firms for size and winners minus losers for momentum). Then again the comparison of these long/short portfolios against a riskless investment (bills) makes sense. But it would be interesting to see outperformance against the broad market too. Especially the tables of outperformance which they have in their book would make morse sense to me.

The authors of the book seems to define the premium in any case as outerperformance over risk-less: but can I compare the outperformance of the broad market (beta one) to the outperformance of winners-loser (beta zero)?

How do we define the factor premium? How do they define the premium?

All factor returns (including "passive" factors like equity premium, credit premium etc) should be assessed using the fairest possible basis for comparison - self-financing portfolios.

For a passive long-only investment, that is equivalent to the total return on the asset class minus the risk-free rate. For a tradable factor premium, that means constructing long-short portfolios to isolate the factor (e.g. buy cheap stocks, sell expensive stocks with the same dollar value).

The assumption is that you can borrow or invest cash at the risk-free rate, and that any self-financing portfolio can be leveraged as much as necessary by borrowing. I'd add a few comments to this -

1. If you are assuming that you can borrow or lend at the risk-free rate, you are free to compare strategies using risk-adjusted returns (e.g. using the Sharpe ratio) because low volatility strategies (e.g. bonds) can be leveraged to the same volatility as high volatility strategies (e.g. equities)
2. In reality you can't borrow or lend at the risk free rate. The spread between what you can borrow at or what you can invest at is the cost of leverage.
3. For some factors, particularly low-volatility, low-beta or defensive factors, it is difficult to construct a self-financing portfolio by being long/short stocks with the same dollar value. This is because you are also trying to construct a zero beta portfolio (to avoid loading on the equity risk premium) which means that you want your long portfolio (low beta stocks) to have a larger dollar value than your short portfolio (high beta stocks).
4. Related to point 2. - the existence of a cost of leverage (either purely financial, or regulatory/risk-aversion) is often cited as the reason for the existence of the low-beta premium in the first place.

It doesn't make much sense to compare a zero beta factor portfolio to the performance of a beta one equity investment, because the strategies are uncorrelated by design. Of course, you can compare them, as you can compare any two things (even apples and oranges) but you don't learn much.

Who cares if a zero beta value strategy is outperformed by a long-only equity investment? These are different strategies, capturing different risk premiums. In different time periods, one or the other will outperform. A beta zero strategy is not designed to provide "equity exposure plus alpha". It is best thought of as an entirely separate return stream, which it is possible to isolate and make an entirely separate allocation to.

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Note that the definition of the equity risk premium -

$$R_{\rm ERP} = R_{\rm Stocks} - R_{\textrm{Treasury Bills}}$$

is not "how much do I earn if I invest in stocks instead of bonds". Rather, it is how much do I earn if I invest in stocks instead of risk-free instruments, where "risk-free instruments" are proxied by Treasury bills.

There is an entirely separate bond risk premium representing the compensation for bearing interest rate risk (and possibly sovereign default risk) which might be written

$$R_{\rm BRP} = R_{\textrm{Treasury Bonds}} - R_{\textrm{Treasury Bills}}$$

where we could use, for example, the return on a constant duration portfolio of 10Y bonds to proxy for the "Treasury Bond" return. You may also here this referred to as the duration risk premium, or the reward for duration extension.

• You are right about the mix up with risk-less and bonds - perferctly right. Do you know the book that I mention? They analyze ouperformance and premium and I don't really understand whether they use the same definition (comparison to risk free) for the market risk premium (beta=1) and factor premia (beta = 0) ... it looks as if they do - which does not make sense (as you agree, right?). – Ric Jun 26 '17 at 14:12
• I don't know the book, but Larry Swedroe has a good reputation. My only point is that for comparisons between portfolios to be meaningful, portfolios must be self-financing (i.e. you are always looking at return in excess of the risk-free rate). Beyond that you can do any comparisons you like, but some comparisons are more useful than others. For example, if someone comes to you with a strategy that is net long equities it makes sense to regress the returns against the market, to see if it has alpha. It doesn't make much sense to do that for a portfolio that is market-neutral by design. – Chris Taylor Jun 26 '17 at 15:18
• here gestaltu.com/2017/01/definitive-book-factor-investing.html they describe the book and the risk-premia tables. They write: "One great feature of the book is that each chapter contains a table describing the percentage of periods where each factor produces positive excess returns (i.e. above Treasury-bills) over 1, 3, 5, 10 and 20-year periods. " To me it looks as if they compare market beta (beta = 1) and factor pfs (long/short thus beta=0) to riskless. Thus they do compare different kinds of fruit. What do you think? – Ric Jul 2 '17 at 11:59