Why does Exchange ActiveSync not keep more than 'x' days of mail? I have a HTC Desire X. I have set Exchange ActiveSync to download past 3 days. However, why does it not keep older mails which are already downloaded? i.e. on Friday, I can see mails from Tuesday on. On Saturdays, I can only from Wednesday on. The Tuesday email which was already downloaded and read is no longer visible? 

Is there a way to make not delete the local copy of older emails?

There is an earlier question which is similar - Why does stock email app only keep 25 latest emails?

However I don't think the accepted answer there is relevant here. From that answer


  to prevent large transfers of data that end up using the device communication thus leading to a faster battery drain;


Why would keeping older mails cause large transfer of data? The older mails are already downloaded and are there on your phone.


  to preserve memory consumption that in turn would leave your device slow and inefficient.


Just because the older mails are there on disk, it will not increase memory consumption. A better way to program it would be display '3' days but have a way to show older ones on demand - since the app would read it from disk only on demand, it wouldn't increase memory consumption.

My phone runs Android 4.0.4.
I don't think this is a setting on your phone or has anything to do with data transfer or memory consumption (at least as far as your device is concerned). I think this is a policy enforced on your Exchange account. Exchange ActiveSync allows account administrators to define policies on how email is delivered to and stored on your device.

Specifically, an email retention policy called Maximum Email Age specifies the maximum number of days of email items to synchronize to mobile devices. According to Microsoft, Exchange 2013 allows the following options for this policy (emphasis added):


  All, OneDay, ThreeDays, OneWeek, TwoWeeks, OneMonth


HTC has a PDF document that outlines the Supported Microsoft Exchange Policies in HTC Sense (Sense is your phone's stock UI) and this policy is supported by your device. It would seem as though your account's administrator has set this policy to only allow three days' worth of email storage. Justifications for this kind of policy vary; I recommend contacting your account administrator (or whoever operates the Exchange system you use).