Moses Christensen, of Corser Street, Stourbridge, stabbed 70-year-old Richard Hall "just for the sake of killing somebody", he later told police.
The body of Mr Hall, who lived in Perton, South Staffordshire, was found on Brown Clee Hill in August last year.
Christensen turned up on a nearby doorstep soaking wet, having tried to hide himself in a pool.
He was interviewed by police and readily admitted to stabbing Mr Hall to death, but later denied murder on the grounds of diminished responsibility.
Christensen's trial was held at Stafford Crown Court, and a jury unanimously found him guilty after deliberating for less than three hours.
He has been warned to expect a life sentence.
The trial heard from psychiatrists who had assessed Christensen, including Dr James Collins, a consultant forensic psychiatrist at the secure Ashworth hospital where Christensen was previously a patient.
The doctor provided evidence to the trial following his own interviews with Christensen and conversations with his family.
Dr Collins painted a picture of a man who had once been a "caring, sensitive and thoughtful boy" but who by the age of 13 had begun experiencing severe paranoia, psychosis and delusions about the people around him – the court heard that he came to believe that other people in his life were "props" in a fake world that had been arranged specifically to make his life miserable.
Coverage of the case:
In conversations with psychiatrists he expressed his desire to kill every other human being on the planet "on an atomic level".
He would research psychopathy and the Columbine High School massacre online.
At the time he met Mr Hall in the Shropshire countryside, he had "nothing but the clothes on his back", the court heard.
Adjourning the case until March 16, Mr Justice Pepperall told Christensen: "I am going to delay sentencing.
"But understand this. There is only one sentence that can be passed and that is a sentence of life imprisonment."