How do I model a dependent variable that is a proportion? Problem
I am trying to test gender differences in the risk propensity of investors related to specific kind of stocks. More specifically, I want to the test the hypothesis predicting that female investors would be less risk averse than male investors. 
Data
For each stock in my dataset, I have riskiness of the stock (continuous variable), along with IDs of individuals who have invested in the stock as well as their gender. Each stock can attract multiple investors and all individuals could potentially invest in multiple stocks.
Model
My thought was to have proportion of female investors for a stock as a dependent variable and riskiness as a predictor. A positive and significant coefficient of riskiness would indicate support for the hypothesis. 
Question
 1. Is this a correct specification of model?
 2. If so, what type of regression should I use?
 3. How do I control for random effects of investors in a mixed model?
 3. How do I implement this is in R?
 A: I would treat riskiness as a continuous outcome and see if there are differences in the average riskiness between males and females. Moreover, it seems that you have a crossed design. Hence, I would start with a model of the form (using function lmer() from the R package lme4):
fm <- lmer(riskiness ~ gender + (1 | ID) + (1 | stock), data = your_data)

The coefficient for gender will denote the difference in the average riskiness between males and females.
In addition, you could also consider whether the differences you may see between males and females are attributed to other factors, e.g., age, education, etc. In this case, you may want to also control for these variables in your model.
A: A popular way to regress proportions to variables is to use the logistic regression.
Before fitting a model, you can also go non parametric and just plot the proportion of investors against the quantiles of riskiness.
A: If you insist on using this as your dependent variable, estimate a censored regression, since the dependent variable is bound on the interval [0,1]. 
As mentioned in the comment above, logistic regression would be interesting, if you have investor-specific (beyond gender). A dummy variable for gender would allow you to look at an individual's probability of investing (though you may not have adequate data for this. It's hard to tell from your description.
Other interesting questions could come from looking at the size of investments, rather than measuring investments as a binary variable. By looking at the proportion of investors, you aren't distinguishing between someone investing one dollar and someone else investing one million dollars.
