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At the end of the 5 years one will have collected $500 in interest, and "given up" $200 in face value, for a net of $300.
I tried to solve for Yield to Maturity in my head, and I get 5%.
5% of $1,200 is $60.
$60/year for five years gives $300.
Which is how it actually works out.

I think the last line is your issue. You don't actually get $60 / year. Think of cash flow which is what dictates the value of any investment.

Time = 0 -- you pay out $1200
Year 1 - 4 -- you get $100
Year 5 -- you get your final payment + $1000

So you get 500 in interest as you said, and then lose $200, but you can think of that as being more than 5% because you get the $500 in cash (on average) before you lose the $200 (which only happens at the end).

Money next year is worth more than money in 5 years, and that is why the YTM (which use an implied time value of money) is >5%.

Hope that makes sense. If you want a thought experiment, think of the following choices:

1) You can spend $1200 to buy the bond you mentioned.
2) You can spend $1200 to buy a 5% coupon $1200 face value bond.

Which would you rather buy and why? YTM is simply trying to make that choice mathematical.

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