# Why are calculations typically performed on excess returns vs returns

I have noticed that most of the research that I read tends to perform all calculations on a series of excess returns over the risk free rate. My question is, “Mathematically why is it necessary to use excess returns rather than just using the raw returns?” For instance, if I am performing a regression with the monthly returns of a mutual fund as my dependent variable and the monthly returns of 4 benchmark indices as my independent variables it seems to me that whether I use the raw return series, or if I subtract out the risk free rate from each return of my dependent and independent variables, I should get roughly the same result. The beta coefficients may be slightly different but relative to each other they should all be the same. So what is the significance of running an analysis on an excess return series over the raw montly returns?

Your R^2 won’t change (unless there is noise in the RFR figures), but your betas will. The betas measure exposures to risk, so you want those to be as accurate as possible, since your performance as a manager is about how effectively you convert risk-taking ability into investment returns.

relative ranking might change depending on the value of risk free rate because Alpha = E®-RFR-beta*(E(m)-RFR). I can post a simple example, if you need a clarification.

I disagree on the concept of “excess returns”, how can return ever be excessive.

Perhaps when the excess return is negative?

Lexcessive return…less than excessive.

I appreciate the responses. I have noticed that most black-litterman type calculations also work on series of excess returns but I believe that is to make sure the return vector is just the excess return over the market and then it is up to you to decide what the market risk premium is based on heuristics or CAPM. So for example: I can take 6 monthly index return series from ibbotson and dump those into excel, lets just say Russell 1000 growth, Russell 1000 value, Russell 2000 Growth, Russell 2000 Value, MSCI EAFE USD, and MSCI EM USD. If my expected investment horizon is 10 years I could take the yield on 10 year treasuries as my risk free asset. Now ibbotson gives you the annual yield so I guess I could just divide each one by 12 to get a monthly yield. Drop that into excel next to my other return series and then subtract each monthly yield on 10 year treasuries from each monthly return on my 6 index return series to get 6 new return series of excess returns. Does this sound right?

jg, your approach sounds reasonable. I would work with daily returns if possible.

Agree with maratikus that it’s reasonable. Some people use 90 day treasuries if you are going to be rebalancing at quarterly or more frequent intervals. In effect, that means your investment horizon is 90 days as opposed to 10 years, but stock generally returns scale over time better than fixed income or derivative returns.

jg, why wouldn’t you use excess returns? Said another way, why would you choose to use raw returns?

Well yeah, phrased that way “excessive returns” sound more attractive than “raw returns”. How about poached returns? How about just make some money.

r=rf+B*(rm-rf) Subtracting rf, you get: excess return for the stock=a+B*excess return on the market The slope shouldn’t change much, but the alpha will be adjusted. There shouldn’t be much difference in terms of MVO of the actual portfolio weights. However, the theoretical reason (if I’m not mistaken) for doing it that way is that the MVO is for allocating between risky assets and not the allocation between risky assets as a whole and the risk-free rate. So you can swap out the risk-free rate, get your risky allocation. Then with those weights you can multiply them by the expected returns and divide that by the expected portfolio variance and risk aversion coefficient to get the risky allocation.

jmh530 Wrote: > There shouldn’t be much difference in terms of MVO > of the actual portfolio weights. Incorrect. There could be very significant changes, especially if the covariance matrix has a high condition number. Intuitively that makes sense because we know that MVO is very sensitive to inputs (especially if the covariance matrix has a high condition number). By switching to excess returns, you subtract rfr from each expected return and the portfolio weight can change significantly. Here is a simple example. You have two assets A with expected return of 1% and volatility very close to 0, and B with return of 4% and volatility of 5%. If you are very-very risk averse portion of A will be huge (close to 100% or 100% depending on correlations). However, if you consider risk free rate of 2%, then weight of A will be 0, or very close to 0 (depending on correlations).

> However, if you consider risk free rate of 2%, then weight of A will be 0, or very close to 0 (depending on correlations). Is this just a specific case of the more general PMPT approach of downside deviations(http://en.wikipedia.org/wiki/Post-modern_portfolio_theory) ? To quote: It has long been recognized that investors typically do not view as risky those returns above the minimum they must earn in order to achieve their investment objectives. They believe that risk has to do with the bad outcomes (i.e., returns below a required target), not the good outcomes (i.e., returns in excess of the target) and that losses weigh more heavily than gains. This view has been noted by researchers in finance, economics and psychology, including Sharpe (1964). “Under certain conditions the MVA can be shown to lead to unsatisfactory predictions of (investor) behavior. Markowitz suggests that a model based on the semivariance would be preferable; in light of the formidable computational problems, however, he bases his (MV) analysis on the mean and the standard deviation[3].”

…this all sounds very technical.

DarienHacker, excess returns were used initially in MPT. You are correct, though, that downside deviations might add value especially if distributions are very skewed.

Anyone got a way of measuring future downside deviations?

bchadwick Wrote: ------------------------------------------------------- > Anyone got a way of measuring future downside > deviations? Very easy! Take future return distribution and perform the calculation! Anyone got a way of calculating future return distribution?

^ I think you take the past return distribution and then just throw cr@p at it and see where it piles up!

bchadwick Wrote: ------------------------------------------------------- > ^ I think you take the past return distribution > and then just throw cr@p at it and see where it > piles up! Nice description of Bayesian methods.