Neither man is suited to life in banking and the humour comes mostly from Psmith's dealings with the bosses they encounter, especially the manager, Mr Bickersdyke. Psmith, a languid Old Etonian with an eyeglass, addresses everyone as Comrade and speaks in a formal, mannered, yet comic tone. He is, of course, the central character in the book, although we see events mainly through Mike's eyes. Psmith reminded me of another Wodehouse character, Jeeves. His attitude to his bosses—at once studiously respectful yet discreetly superior—also recalls that of Jeeves to his employer Bertie Wooster. Psmith's distress at the sartorial indiscretions of a young employee in his department is yet another echo of Jeeves.
The story has an autobiographical element. From Wikipedia I learn that Wodehouse, like his two main characters, was compelled as a young man to work, very unwillingly, in the London branch of a Far East bank (both he and Mike had fathers who had suffered financial losses which required their sons to take this course).
In spite of its age the book stands up well to a modern reading; the humour is timeless. It won't disappoint anyone who loves the mature Wodehouse oeuvre.