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Yesterday, I did a redemption for a flight on American Airlines using AAdvanatge points. The flight costs about $800 when booked normally though AA or aggregators like Expedia (tried several and there was a price difference of no more than $12). After redeeming my points for the flight, I still had to pay over $260, which is close to a third of the price! Last time I got an award flight, it cost me $240 for a $500 flight, so almost half. All these were round-trips, if that makes a difference.

Is there any way to avoid or at least reduce the amount to pay when redeeming a flight using frequent-flyer points?

Would redeeming across airlines help? Like using AAdvantage points to book an another One World member. Do some type of flights (one-way, multi-city) or some origins or destinations incur fewer of these fees?

For Bonus Points: What are these charges actually? I assume airport taxes but the receipt I got does not specify a breakdown.

With American, you can't avoid the fees completely, it says so in their terms.

AAdvantage flight awards are subject to, and the passenger is
responsible for, any applicable departure taxes, security fees,
federal inspection fees, passenger facility charges and/or other
taxes, fees and surcharges assessed by appropriate authorities or
partner carriers. In addition, there may be applicable non-refundable
charges under the AAdvantage program, such as AAdvantage award charges
or upgrade award co-pays.

However, you can reduce those fees in at least two ways.

1- Book entirely on American. This avoids fuel surcharge that other partners such as BA do charge you

2- Avoid international travel. I assume, given your previous posts, you're flying out of Canada. We have expensive airport fees and this hikes the total American charges you.

Disclaimer: I work at Flightfox

To understand how airlines calculate fees and taxes on an award ticket, first you need to understand what makes the price of a cash ticket.

Let's use a one-way AA ticket from NYC to London an example:

Total price of the ticket is $1,498.90 and it consists of:

1. 
   Basefare (Fare 1: AA H1N0O1C5 NY to LON) — $1,212.00
   


2. 
   Fuel surcharge (AA YR Surcharge) — $259
   


3. 
   Local and government taxes (Departure tax, Security Fee, Passenger Facility Charge) — $27.90
   

A base fare is what the airline actually charges for the ticket. In plain words, it's the money that the airline makes selling you a ticket.

A fuel surcharge was originally introduced for airlines to easily change the airfare prices in face of rapidly changing oil prices. Currently, airlines use fuel surcharges in several different ways. One of them is to charge extra money for award (miles) flights.

Taxes are paid by the airlines directly to the government or airport.

What do airlines charge for award (miles) flights?

When booking an award (miles) flight, you are never charge the base fare. What you can be charge is fuel surcharge and taxes. For example, any domestic award ticket using AA-only flights will cost you $5.60 total (September 11th Security Fee). However, a ticket on British Airways (eg. to London) will cost you several hundred dollars because the fuel surcharge will be added to the total price.

You can check the fare structure of a flight that you want to book with miles by using ITA Matrix

How to lower the amount of fees and taxes on an award (miles) ticket?

1. Use your miles for airlines that impose no or low fuel surcharges. For AAdvantage miles, try AA or Cathay Pacific and avoid British Airways.


2. Use miles for trips originating from countries with low taxes (eg. USA), avoid tickets from/to Great Britain ($95 in economy and $190 in business class extra tax)


3. Don't book last minute as many airlines add late-ticketing fees (AA: $75 if booked within 21 days of departure)

When is it a good deal to book with miles vs. cash?

I recommend to use your miles when 10,000 miles AAdvantage miles save you at least $100 dollars, but preferably $150. The best way to use your miles is for business and first class flights, where 10,000 miles can save you $1,000.

Call AA back and they will tell you all the fees involved.

Based on the amount, this sounds like an international itinerary so there's piles of Government mandated fees on both ends which they would still have to pay or are categorized as user fees which you still have to pay. The airline is just collecting and remitting it on your behalf.

You are using miles for the fare but the Governments still want their money.

You can ask if there are any waiveable fees but do not expect any.