Calvocoressi was born in Calcutta, the only child of Matthew John Calvocoressi (1873–1939). His father was descended from a Greek family from Chios; two uncles married into the Ralli family. He was a director of the Indian branch of Greek merchant business, Ralli Brothers. His mother, Agnes Hermione Melville, was of Anglo-Scottish ancestry and was the granddaughter of Michael Linning Melville. His parents moved to London in 1922. He was educated at Eton College and read Modern Languages at Magdalen College, Oxford.[2]
After the outbreak of the Second World War, he was commissioned into the Scots Guards in 1940,[3] and was posted to Egypt with the 2nd Battalion in 1941. He won an immediate Military Cross in 1942,[4] while serving as a lieutenant in command of a platoon of six-pounder anti-tank guns. The battalion was defending a ridge at Bir el Rigel in Libya. After days of intense fighting, the battalion was attacked by two armoured columns from the 21st Panzer Division on 13 June 1942. His platoon destroyed five German tanks before it was overrun. Calvocoressi was captured, but escaped the following night. He rejoined his unit the next day after walking 17 miles (27 km) across the desert.[2][5]
He became a stockbroker after the war, concentrating on private client work. He moved to the town of Westerham in Kent in 1950, near the country home of Winston Churchill at Chartwell. He was instrumental in the erection of a statue of Churchill by Oscar Nemon in the town in 1965.[2] He was High Sheriff of Kent in 1978–79.[6]
He was survived by his wife, Katherine Kennedy (the sister of Sir Ludovic Kennedy),[5] whom he married on 29 April 1947. They had three sons and a daughter.[2] One son is art historian Richard Calvocoressi.