When the president ends their term of office in several countries, they receive a pension for life. Why is this? (One guess I have is that it could possibly be to secure their future so their presidential decisions are not affected by eventual future financial troubles.)
What were the problems when the president didn't receive a pension?
At least in the US, where a presidential pension was only instituted in 1958, it was seen as embarrassing for a former president to be struggling financially. Per Wikipedia:
Upon leaving the presidency [in 1953], [Harry S.] Truman ... taught occasional courses at universities, including Yale, where he was a Chubb Fellow visiting lecturer in 1958. In 1962, Truman was a visiting lecturer at Canisius College. As a former president, Truman decided that he did not wish to be on any corporate payroll, believing that taking advantage of such financial opportunities would diminish the integrity of the nation's highest office. He also turned down numerous offers for commercial endorsements.
Once Truman left the White House, his only income was his old army pension: $112.56 per month (equivalent to $1,089 in 2020). Former members of Congress and the federal courts received a federal retirement package; President Truman himself ensured that former servants of the executive branch of government received similar support. In 1953, however, there was no such benefit package for former presidents, and he received no pension for his Senate service.
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In 1958, Congress passed the Former Presidents Act, offering a $25,000 yearly pension to each former president, and it is likely that Truman's claim to be in difficult financial straits played a role in the law's enactment.
The current presidential salary is $400,000 (but, it's complicated) and the pension is $219,000 annually (ibid.), as well as many other benefits like office space, treatment at military hospitals, and Secret Service protection.
The details depend on the country, of course.
In Germany, a former president receives not just a pension (called honorary / honor pay), but also some staff, offices, etc. for what are called "remaining tasks" from the office. Basically, the former president is expected to act in a manner consistent with the dignity of the office and the interests of the nation. They tend to be older, with the presidency coming at the end of a long career, which makes them more of elder statesmen than active politicians.
In the United States, Jimmy Carter was one of the more active ex-presidents in recent times. He got involved with North Korea and the Middle East as a credible spokesperson not directly connected to the then-current administration. This allowed him to visit places and say things an active government member might not have said.
Speaking generally, one reason which would make this a beneficial feature of a political system could be to make it less important for the president to be afraid about destitution after their term; and thus make them less susceptible to corruption.
For example, after her 12-year period in office, Angela Merkel will receive 65% of her compensation until the end of her life, which is the maximum amount possible for someone in her role. In her case that's roughly 15.000€ monthly. This means that at no point during her active time as chancellor was she ever forced to care much about what she'd be doing after her mandate was over, and presuming good character, there was little monetary, objective reason for her to bow to any less-than-decent offers during the time when even small decisions would possibly have made huge amounts for some company or other party. Or to jockey for some other position within some company or the political system after their prestigious position is over...
The United Kingdom only instituted an ex-Prime Minister's allowance in 1991 and it is still limited to 50% of salary. There has been a lot of unfavorable press directed at a couple of recent PMs (Tony Blair & David Cameron) and their post-office "business consultancies" where they are clearly aiming for large-scale personal profit.
At least one columnist in the New Statesman feels that a larger pension & continued govt support might have prevented at least some of this and given the opportunity for them to remain statesmen with a higher purpose.
There are many reasons, but they boil down to a few most relevant categories
Respect - Attaining the most prestigious job on the planet arguably should automatically come with financial security.
Image - Imagine the scenario of a former president having to sell personal possessions to keep bill collectors at bay, or being for-closed on.
Security - Keeps him/her less tempted to write some book divulging secrets or from having to work for a shady arms dealer who does business in the Middle East.
Employment options limited - Most come from senatorial or other elected positions that they can't return to.
Tradition - This was decided when being President did not automatically guarantee millions in speaking deals
