What, if any, objective tests were designed for the assessment of altruism?
The following scale; The altruistic personality and
the self-report altruism scale, a commonly used measure of altruistic tendency in psychological research. It has 745 citations at this time November 2017. The scale is also freely available online.

Rushton, J.P., Chrisjohn, R.D., Fekken, G.C., 1981. The altruistic personality and
the self-report altruism scale. Personality and Individual Differences 7,
293–302.

This has been discussed as psychometrically adequate for measuring altruism, and has good the SRAS has good inter-rater reliability. However the SRAS has two main weaknesses: its expansive, in terms of its definition of altruism, it also it frames of reference are unspecified, and unclear.   

There are other measures that investigate aspects of altruism such as the Internalised subscale of the Self-importance of Moral Identity Scale (ISMIS). In the following article they discuss moral behaviour to outgroups, which the scale measures. This is also a highly cited source for measuring altruism or moral behaviour. 

There are a few other scales or methods of measuring altruism, I do not have time to go into detail...

Empathic Concern questionaire (EC)

Compassionate love for others

Sprecher, S. &amp; Fehr, B. (2005). Compassionate love for close others and humanity. Journal of Social and Personal Relationships, 22, 629-651.

State Based Empathy. Empathy is more of a measure of motivation to act kindly which is a precursor to altruism but not actually altruism itself. 

I highly recommend you read the following blog on measuring altruism, which discusses it in far greater detail than the answer here.

T. Farsides. (2014, October 18). Conception and measurement of altruism. Retrieved from http://tomfarsides.blogspot.com/2014/10/conceptualisation-and-measurement-of.html