The Spain Under-21 international left los Blancos just 12 months ago, but he is now back in the Spanish capital after a fine season in Germany
The Spain Under-21 international left the Santiago Bernabeu for Germany last summer, with Leverkusen agreeing to pay €5 million for the defender.
However, Madrid included a buy-back clause in the deal that they have now enacted to bring Carvajal back to the Spanish capital, for a reported €6.5m.
"Your case will become the example for many academy players that the dream can come true," Perez enthused.
"You start a new phase of your career under the orders of [coach Carlo] Ancelotti and you also come here as a European Under-21 champion.
"You have earned the right to be here and know better than anyone what it means to defend this badge. It is a great responsibility but also something wonderful because of what it stands for. Welcome home."
Carvajal, for his part, was just as enthused by his return to Madrid, just 12 months after departing.
"I want to say thank you for the trust that has been placed in me," the 21-year-old stated at his official unveiling on Friday. "Thanks to the decision to go away for a year I can be here with you.
"In 2002 I put on this shirt and began to fight for a dream that has come true today. I am the happiest person in the world. I'll give my last breath for the good of this team and club."
Carvajal contributed eight assists and one goal during a fantastic debut season for Bayer Leverkusen.