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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.

Ben