They found some gold in their house that belonged to a relative, who came back a month later to take it but they had lost it. What do they have to do?
Question

                Once a relative visited us when he left (or forgot) a piece of gold in our home without telling us. On finding that gold, we notified him to come back to take it. But he didn't come immediately, as he waited a whole month till he appeared again, during that time we have lost that piece of gold which was covered by an old paper. My father is an old man with a poor memory so maybe he has thrown this folded gold here or there not knowing it's a gold. Now, we think of paying him back the half of estimation of that piece of gold price (as recompense) but nowadays the price of gold is doubled so paying a half equals paying all amount. question is must we give our relative a payment valued by gold price according to the time of buying (when our relative did buy that gold in the past) or according to the time of (current) selling (exactly when we have lost it)?.
Praise be to Allah.
      Firstly: 

    This gold that you found was a trust (amaanah) that was in 
    your care and that you had to take care of until its owner came to take it. 
    The ruling concerning a trust, if it is lost by the one in whose care it was 
    due to some shortcomings or negligence on his part is that he is liable for 
    it and must compensate its owner; if its loss was not due to any 
    shortcomings or negligence on his part then he is not liable and is not 
    obliged to do anything. 

    Ibn Qudaamah (may Allah have mercy on him) said: 

    Any item left in a person’s care is a trust (amaanah). If it 
    is lost or destroyed without any negligence on the part of the one to whom 
    it was entrusted, he is not liable regardless of whether any of the property 
    of the trustee was lost along with it or not. This is the view of the 
    majority of scholars.…

    But if the trustee neglected it or fell short in taking care 
    of it and it was lost or destroyed, then he is liable and there is no 
    difference of scholarly opinion that we know of, because the one who 
    destroys the property of another is liable for it, as is also the case if he 
    destroys it without it having been entrusted to his care. End quote. 

    al-Mughni, 7/280 

    So you should have kept this gold in the same safe place 
    where you put your own gold, and you should not have put it with papers that 
    were easily accessible to people. 

    Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have 
    mercy on him) said: 

    Storing gold and silver is not like storing pots and pans. 
    Pots and pans are kept in clear view in the house and are not locked up, but 
    gold and silver are kept in boxes and chests and are locked away. End 
    quote. 

    Al-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘, 
    10/288 

    What appears to be the case is that you fell short in taking 
    care of this gold, so you are liable for it. 

    The basic principle with regard to liability is that you are 
    liable for replacing it with something similar to what it was on the day it 
    was lost. If it is not possible to replace it with something similar, then 
    you have to pay its value. If there is a piece of gold on the market that is 
    similar to the piece of gold that belonged to your relative, then he should 
    be given the choice of either having you buy it for him or you giving him 
    its price. 

    If there is no similar piece of gold in the market, then you 
    have to work out the value of his piece of gold at the time it was lost, 
    then give him its price. 

    Shaykh al-‘Uthaymeen (may Allah have mercy on him) said: 

    What counts [with working out the value of a lost or 
    destroyed item] is the time when it was lost or destroyed, because that is 
    when it ceased to be in the possession of its owner, i.e., at the time when 
    it was lost or destroyed.

    End quote. 

    Al-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘, 
    10/122 

    This is what you have to do for this relative, unless he 
    waives his rights in full or in part. 

    And Allah knows best.