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YTM can be determined using the price formula of a bond.
Part a
The present value (price) formula for a zero coupon bond is:
PV = F/(1+i)n
Plugging in the given information,
$300 = $1000/(1+i)15
(1+i)15 = 3.33
1+i = 1.08357
i = YTM = 0.08357 = 8.36%
Thus the YTM of the fifteen-yr zero coupon bond is 8.36%.
Part b
Semiannual coupon = 6%*$500/2 = $15
Price = $15/(1+(i/2)) + $15/(1+(i/2))2 +..... + $15/(1+(i/2)6 + $500/(1+(i/2)6 = $400
On trial and error using Excel, YTM has been determined to be 7.23%.
Thus the YTM of the three-yr semiannually 6% coupon bond is 14.45%.
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