# Is there a way to meaningfully generate daily returns from monthly?

I have a set of 7 investments in a portfolio and I need to optimize the weightings based on some exposures to various markets/styles/economic factors. I was hoping to do some sort of simple exposure analysis or 'factor analysis' (not actual Factor Analysis, but more just a bunch of regressions), using daily returns of various risk factors (for example, SPX, TBills, MSCI, FamaFrench Factors, etc).

I only have daily returns for 5 of the 7 investments in the portfolio. I have monthly returns for the remaining two. Is there an easy way to do some sort of generation of daily returns from monthly returns, possibly modelling the monthly against the factors' monthly returns, and then generating daily returns based on the model? (I know this is circular, but I am spitballing.) The problem is that I need some way to tie or anchor the modeled daily returns back to the actual monthly returns.

Any ideas? And does this make sense?

• Have you looked at using brownian bridges for each if the periods? I've had a reasonable amount to drink, but it seems to be exactly what you want...
– will
Commented May 29, 2016 at 0:42
• Great Question! Commented Jul 26, 2016 at 13:55

This is a commonly seen problem, and also relates to situations in which one is dealing with some less-liquid underlyings. I will describe a method that you could think of as "stochastic backfilling" - it uses the "Correlated Brownian Bridge" technique. There are many references for the standard BB technique on the web.

Let us assume we have one data series which is daily and one weekly, and that the time series are correlated.

How can we backfill the "missing" data from the weekly series in a way that is consistent with the overall distribution shape of the weekly data series, and also maintains the correlation with the daily time series?

This backfilling will be adding no new information, and there will be uncertainty around the result, as we will effectively be doing a correlated MC Brownian Bridge simulation to fill in the "missing" data.

The first step is to model the distribution of the weekly time series. Here there are a lot of choices possible. Let us assume we model the time series using a Student t distribution and do an MLE fit. We will use this information to generate fictitious returns with the right distribution.

The second step is the model of the copula that connects the two distributions. For simplicity let us assume a Gaussian copula, but there may be better choices.

We now set up a two factor MC simulation, except that the daily factor will have a known return, and the weekly factor will have randomly generated returns. The simulation will run for the five days for a given week. I will not describe how to write a two factor MC simulation here, and assume it is known.

Having generated the "first cut" at the five individual days' returns for the weekly time series, we see than in general they will not "compound up" to the actual observed weekly return. In this case we adjust the returns by adding a constant drift.

The simple way to think of the drift is to imagine a weekly price that goes from 100 to 110 over a week. Let's say our simulated prices go from 100 to 120 over that week. To make the time series "reconnect", we simply subtract a constant daily drift from each generated return that forces the final price to be 110 instead of 120.

Of course, each MC run will produce a different Brownian Bridge between the two weekly returns. This is normal and represents the uncertainty around having backfilled this information: we have not created any information here. A number of simulations can be run and the uncertainty noted. There will already be uncertainty in your regressions, and this will add to it.

And of course this method can be extended to not just two underlyings but any number, thereby "backfilling" missing data in a way that is consistent with both the observed probability distribution and with the correlation structure of the data.

Kalman filter (or similar methods) are quite well suited to deal with observations that are of different sampling frequencies and/or asynchronous.

There are a few ways to do this. For example, the FRBNY (google FRBNY and nowcasting) creates a weekly GDP number from monthly and weekly series. You can sift through that to see how they change the time steps. In the past I generated weekly unemployment data (which is a monthly series) from the pattern of weekly unemployment claims or something like that, it was a long time ago. Just make sure the changes you estimate get you back to the original monthly data. To do daily, I could have fit these weekly to an Nth degree polynomial. I know that sounds ad-hoc but as long as you know your weaknesses in the model, you can allow for that and judge it appropriately. I used the economic series example cause you mentioned economic indicators, so if your missing series is a financial one there could be a much better fit and approach if there is a similar daily series already.

• "Nth degree polynomial" - Interesting approach @horseless. Have you tried other (linear interpolation, gaussian regression) as an alternative? Commented May 26, 2016 at 23:16