Perhaps God permitted the loss of this rosary so that you may grow in the virtue of detachment? Whether or not that's the case, I hope you find it.
THIS^^^. xavier, given how you characterize the beads as both your favorite and also being expensive, perhaps the intent here is that you be more concerned about the PRAYER of the Rosary, the spiritual and immaterial aspect, than in the material aspect of the beads.
We also have to be aware of how prayer "works". One of the best descriptions of prayer I've read was from a Church Father (or some early monk or hermit ... can't quite recall). In any case, he gave this vivid metaphor for prayer. There's a man in a boat at the middle of a lake, fishing. At one point, he snags something. So he starts to reel it in. He's struggling mightily with it, and figures it must be quite the catch since it's requiring all the strength and effort he can muster. Gradually it starts getting closer and closer to his boat, and JUST when he's about to pull his catch onto the boat, he realizes that he's actually snagged a huge rock. So the entire time he thought he was pulling in the catch and drawing it closer to the boat, he was actually drawing the boat closer to the rock. Similarly, when we pray, we are not somehow bending God's will to ours, getting God to "change His Mind" as it were. Instead of bending God's will to ours, we are actually bending our wills to God's.
God's will for us is always that which is best for us. Since the formal intent of all prayer is actually for our good, God is always answering every prayer, formally speaking, even if He does not materially answer it. Whenever we petition anything for ourselves from God, it's always with the intent of increasing our happiness, and God knows that which would increase our REAL long-term happiness, even if sometimes it means sorrow in the short term. So He always answers our prayer with a view toward our ultimate happiness, thus answering the formal intent of all prayers of petition. When we ask for anything, whether it's for material things, or money, or good health, or ... anything, really, we would like the money because (we think) it'll make us happy. We don't actually want the filthy lucre, the dirty paper bills, ... or even the shiny metal. We want the happiness that we think would come with it. So the formal intent of our prayers of petition are really ... our happiness. And those other things we ask for are just means to that end (or so we think).
It's similar to a child who asks a parent for some candy. Sometimes the parent obliges. But sometimes the parent decides that it's not really for the child's good and ultimate happiness, as too much candy can cause ill health, perahps diabetes or other ill effects, so that while a child might be pleased in the short term with candy and displeased with not receiving it, the temporary sadness and displeasure will be outweighed by the fact that the parent is preventing the child from a life of unhappiness due to ill health ... the misery of diabetes, cancer, heart disease, or whatever. So the formal intention of the child's prayer is happiness. That's really what he wants, and the candy is just a means to that end. He believes the candy will make him happy. But he doesn't have the experience or wisdom to know that it might make him very unhappy down the road. Perhaps he'd end up fat and therefore miss out on courting someone for marriage who might otherwise have been interested in him, and would have made him happy ... but instead would end up alone ad miserable, etc. You get the idea. Of course, we parents just take guesses. "Well, I think this candy might lead to some painful drilling at the dentist's office, or even diabetes, cancer, obesity and heart disease."
But God does't guess. God absolutely knows what the outcome of each request will be. And He can see every possible sequence of causes and events, all infinite possibilities. I myself have experienced a few situations in life where I could not possibly see how some great adversity benefitted me. But, then, in some cases 10 or 20 years later, suddenly the light dawned on me. "Ah, NOW I see why this, that, or the other thing happened to me all those years ago." But sometimes we'll never understand, in this life. It could be a series of 50 things that might have happened as a result of one incident that would never have come about if one little thing didn't happen first. We simply have to TRUST God. We KNOW that He answers EVERY prayer. We never question when He has refused to materially answer some of our prayers in the terms that we wanted. We might think that A will make us happy, but God knows that A will make us unhappy (longer term), whereas B might actually make us more happy.
If we recall Our Lord's teaching about -- What father, if his son asked him for bread, would give him a stone, or if his son asked for a fish, would give him a sake? So the Father will neither give us something useless (i.e. a stone) or, worse, something that's harmful (a snake). But, similarly, what father would, if his son asked him for a poisonous snake, give him the poisonous snake? We would disallow it and would not comply with the request. But we do comply with the formal intent of the request, the true happiness of the child, by preventing him from dying of a poisonous snake bite. I know that when sometimes my children ask for something and I have to refuse for their own good, it pains me to see them unhappy, but I know that by complying, I would be harming the child. So if we, wicked as we are, act this way toward our children, how much more so Our Father in Heaven.
So, rest assured, xavier, both God and St. Anthony have heard your prayer and have answered it. You have the opportunity for great merit by responding, "Thank you, Father, and thank you, St. Anthony, for not finding my Rosary for me. I know that it's somehow for my good and for my benefit."
And by making such a prayer of resignation you gain infinitely more than you would from the physical possession of the rosary, the graces received from which, if you realized how much you gained from it, you would prefer never to see this particular rosary again than to have not had the opportunity to receive these graces.