A monarchy in whatever form provides the stability to a country that democracy in many situations is unable to provide. In a monarchy the successor is bred from birth to fulfill his or her position rather than with an elected head of state who may have considerably less experience. In a republic (a country without a monarch) the head of state owes his power to the private interests that got him elected, and thus becomes indebted to them once he or she assumes power.
In this way the president must satisfy those special interest groups first before he can act for what would be best for his or her country. In a monarchy however the monarch is king or queen simply because he or she was born into the role, and thus doesnt owe anyone or group for where he or she is, thus allowing the monarch to reign more freely and make decisions that are best for the country rather than the group of rich and powerful people that would have gotten him/her there as in a republic. Because the monarch reigns for life there becomes a greater amount of pressure on the monarch to rule as best as he/she can and so that things won’t be messed up for the next generation.
In a republic however, the presidents term is often short, and whatever longterm problems or screw-ups that occurred in one presidents term become carried over to the next man in office who may or may not deal with them, potentially creating a snowball effect of presidents deferring difficult decisions to later presidents (such as the immense debt problem in the US).
A monarchy is ultimately cost saving because the personal fortunes of the royal family can usually cover the costs that incur for the institutions support.
In the UK for example, the monarchy is paid for by a tax (which is like one cent per person per year) for the monarchy’s upkeep. In turn however the private income from the Queen’s estates is turned over to the treasury which in total becomes a net gain for the national treasury (see the link about the civil list). In addition the amount of money gained from tourism brings added revenue for the state. In a republic the salary of the president is paid for by the state, and the private income of the president is kept by the president.
After the president leaves office, his support is still born by the state in the form of pensions which are paid until his or her death. This can be very costly when combined with the dozens of other ex presidents living off of pensions from the state.
Because a monarch is not apart of any one political party he or she is able to be above the factions of politics and make decisions based on his or her conscious not based on what his or her party policy is. The monarch would also be able to mediate between the different levels of government should any emergency arise.
Monarchy is not necessarily backwards and republics are not necessarily better.
Some of the most stable and prosperous nations (Denmark, Norway, The Netherlands, Sweden the UK) are monarchies.
Republics are not more modern and monarchy is not old fashioned.
Anyone with the simplest knowledge of history knows that the ancient world had both monarchical forms of government and republican ones.