In boxing, there are weight classes so that boxers never “punch above their weight.” In a street fight, however, a little man must “punch above his weight” to defeat a bigger man. The boxing metaphor has almost exclusively been used in politics.
 
British Foreign Secretary Douglas Hurd is frequently given credit for saying that Great Britain “punches above its weight” (or “punches beyond its weight”) in international affairs, said in the speech “The New Disorder” at Chatham House, London, on January 27, 1993. However, in October 1990, Deputy Prime Minister Sir Geoffrey Howe said, “If we in Britain are to punch beyond our weight, we have to be part of a team.”
 
U.S. President Barack Obama has often used the metaphor. In March 2012, a Danish TV show aired clips of Obama saying individually to Denmark, then to Norway, then to the Netherlands that each nation punches above its weight in international affairs.
 
   
BBC News
UK’s world role:
Punching above our weight
One of the recurring themes of Foreign Policy over the years has been the notion that Britain “punches above its weight”. This boxing metaphor says much about Britain’s sense of identity: national pride is tinged with a suspicion that we don’t quite deserve our place at the top table of world affairs.
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The phrase “punching above our weight” was coined by Douglas Hurd during a lecture at Chatham House - the headquarters of the Royal Institute for International Affairs and one of the most respected of the Foreign Policy think tanks.
 
Wikipedia: Geoffrey Howe
Richard Edward Geoffrey Howe, Baron Howe of Aberavon, CH, QC, PC (born 20 December 1926) is a former British Conservative politician. He was Margaret Thatcher’s longest-serving Cabinet minister, successively holding the posts of Chancellor of the Exchequer, Foreign Secretary, and finally Leader of the House of Commons and Deputy Prime Minister. His resignation on 1 November 1990 is widely considered to have precipitated Thatcher’s own downfall three weeks later.
 
Google News Archive
12 October 1990, Spokane (WA) Chronicle,  “Discord emerges at conservatives’ party gathering” by Maureen Johnson (Associated Press), pg. A3, col. 2:
“If we in Britain are to punch beyond our weight, we have to be part of a team,” declared Deputy Prime Minister Sir Geoffrey Howe in a pointed remark.
   
The Independent (UK)
Statesmanship must have something to sell
ANDREW MARR Thursday 04 February 1993
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The assumption that Britain’s military posture enables it to ‘punch beyond its weight’ in diplomatic and commercial terms deserves to be challenged. In his recent speech at Chatham House, Douglas Hurd argued that, since trading countries such as Britain prospered in a peaceful world, our enthusiasm for UN peace-keeping was on all fours with a cold, clear-eyed assertion of national interests
 
Google Books
The Making and Remaking of the British Constitution
By Michael Patrick Nolan and Stephen Sedley
London: Blackstone Press
1997
Pg. 6:
... though it may well be true, in a phrase often attributed to Douglas Hurd, that we ‘punch above our weight’.
 
21 March 1997, Toronto (Ontario) Star, “Cuts endanger our spot on world stage” by George Barthos, pg. A27:
CANADIANS ARE forever being invited to “punch beyond our weight” on the world scene, and we’re usually happy to try.
 
Google News Archive
5 February 1998, Victoria (TX) Advocate, “That special relationship” (editorial), pg. 8A, col. 1:
As a military force, Britain proved during the Persian Gulf War that, in the words of former Foreign Secretary Douglas Hurd, it still “punches above its weight” and remains America’s most reliable and most effective ally.
   
The Economist (UK)
Blair of Arabia
Apr 23rd 1998
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Lady Thatcher and John Major seldom crossed the Americans either. But both seemed to harbour delusions about how much Britain could “punch above its weight” in foreign affairs.
   
Google Books
Unfinished Business:
India in the World Economy
By Deepak Lal
New York, NY: Oxford University Press
1999
Pg. 196:
It is because, as former UK Secretary of State Douglas Hurd put it, being a nuclear power and a member of the Security Council allows the UK ‘to punch beyond its weight’ in the international arena.
   
Google News Archive
2 April 2000, Daily Union (Junction City,KS), “Cuban-Americans flex their political muscle,” pg. 3, col. 2:
MIAMI (AP)—In boxing terms, South Florida’s Cuban-American community punches above its weight, especially in a presidential election year.
 
11 September 2006, Brandon (Manitoba) Sun, “Just watch me, says PM” (editorial), pg. A4, col. 2:
The prime minister is punching above his minority government weight class, but he’s right to do so on this one—the Liberals or any other party would be stupid, indeed, to battle the Conservatives in the next election over attempts to reform the Senate.
 
The Weekly Standard
Danish TV Host Mocks Obama for His Rhetoric
“Maybe the copy key got stuck on the presidential speechwriter’s keyboard.”
4:23 PM, Mar 23, 2012 • By DANIEL HALPER
Thomas Buch-Andersen, host of the Danish TV show Detektor, mocked President Obama’s political rhetoric in a recent episode. “Obama used a metaphor from boxing to explain Denmark’s role in the world,” says Buch-Andersen, introducing the segment.
 
He then rolls the tape. “That’s fairly typical of the way that Danes have punched above their weight in international affairs,” President Obama says at a press availability in the Oval Office with Prime Minister Helle Thorning-Schmidt of Denmark.
 
“It’s nice to be praised,” Buch-Andersen remarks. “We punch harder than our weight class would suggest. But how much should we read into his words? According to Obama, are we doing any better than, say, the Norwegians?”
 
The TV host again turns to the tape, this time showing President Obama in the Oval Office with Norwegian prime minister Jens Stoltenberg. “I’ve said this before, but I want to repeat: Norway punches above its weight,” Obama says.
 
Back to Buch-Andersen. “Hmm. So Norway packs a punch too. But what about the Netherlands? Here, their head of government, Mark Rutte, visits Obama.”
 
The tape rolls yet again. “We have no stronger ally than the Netherlands,” says Obama. “They consistently punch above their weight.”