# Estimating moments of annual return distribution

I am interested in estimating moments of annual return distributions of indices with limited historical data. I.e. equity/fixed income indices which have only 10-20 years of daily data available. The ultimate goal is to fit a distribution for sampling annual returns in a Monte Carlo simulation.

What is the standard approach in this situation? Does one really only take the 10 to 20 data points or would one calculate rolling annual returns and then estimate the distribution based on the high number of daily annual returns? The problem with this approach would be a possible dependence between the returns.

Using square root scaled monthly returns doesn't seem correct to me as it would rely already on the assumption of a normal distribution.

Are there alternative methods, i.e. bootstrapping for this kind of application?

• Here a possible dependence is actually dependence. – Richard Hardy Feb 7 at 7:29
• I'm not an expert in this field, but how about modeling your data using a small number of parameters (e.g. linear regression after appropriate transformation) and then bootstrapping from the estimated distribution? – inmybrain Feb 13 at 7:59
• I would agree with @inmybrain . Since you probably want to sample annual returns a few years in the future you could fit a simple linear regression model (I am thinking $$\text{log return}_t=\beta_0 + \beta_1 \text{log return}_{t-1} + \epsilon_i$$ and possibly model your residuals as something with fat tails such as a Laplace distribution). Then you can pull a few samples sequentially to get a sample path of annual returns. – David Veitch Feb 18 at 15:52
• Thanks for your comments. I will try to model it in this way, too. – p.vitzliputzli Feb 19 at 17:00

## 1 Answer

I cannot guarantee that this is a ''standard approach''. However, in my opinion it is a sensible way of modeling this.

First approach: Model annual log returns directly

You could, in principle, take your 10-20 data points and fit a distribution to them. I would suggest a Bayesian model as this will give you a nice way to quantify the uncertainty arising from few available observations. However, in my opinion, there's no need to throw away so many data points (in case they are available). Note that the annual log return is just the sum over the daily log returns within that year. Hence, you could also opt for a

Second approach: Model daily log returns and infer annual log returns

Using the fact that the sum over the daily log returns gives you the annual log return, it is a better way forward to directly model daily log returns. Hence, you want to fit a distribution to the daily log returns. Many people will choose a normal distribution. However, it has been shown that financial returns often exhibit fatter tails. Hence, I would suggest a t-distribution or a Laplace-Distribution.

You can either directly estimate the moments of these distributions or assume a zero mean (which will be a sensible thing to do for most daily asset returns time series) and only estimate the remaining moments of these distributions (i.e. variances). You can then simply take 365 samples from your fitted distribution of log returns, add them up and this will be equal to one draw from your annual return distribution.

Note that all of the above assumes that there is no time dependence between today and tomorrow. As stated above, in my opinion a zero mean assumption for log returns won't hurt too bad. However, in case you want to do a full and proper time series analysis, you could for instance fit an ARIMA model to the log returns after choosing a proper ARIMA structure using model selection criteria and simulate 365 data points from this model. Again, adding them up will give you your annual log return.

However, if you assume normally distributed errors, the tails of the errors will likely be too thin to properly capture volatility in financial return data. Depending on how far you want to take this, you could also opt for more complicated models. At least in a Bayesian setting it's not too much of a hassle to fit e.g. an $$AR(p)$$ process with t-distributed errors to combine the benefits of time series analysis and fat tailed distributions. It all basically boils down to why you need to model these returns and how much effort you want to put in the model. I hope that the points I mentioned help you find a good solution for you.

• Thanks for your answer. I will test it with monthly returns fitted to different distributions. – p.vitzliputzli Feb 19 at 17:03