I'm having trouble understanding this example in Kellison's Theory of interest: Consider a 100 par value 4% bond with semiannual coupons callable at 109 on any coupon date starting 5 years after the issue for the next 5 years, at 104.50 starting 10 years after issue for the next 5 years, and maturing at 1000 at the end of 15 years. Find the highest price which an investor can pay and still be certain of a yield of: i) 5% convertible semiannually
ii) 3% convertible semiannually
For i) since the bond is selling at a discount it makes sense to me to take the latest redemption date possible which gives the price of 100+(2-2.50)$a_{30|0.025}$= 100-(.50)(20.9303)= $89.53
for ii) it says If the yield rate is to be 3%, then it is not immediately clear which redemption date is least favourable, since the bond will be selling at a premium. Since it's selling at a premium it takes the earliest redemption date in each interval. The prices to compare are:
109.00(2-.3650)$a_{10|0.015}$= $112.37
104.50(2-1.5675)$a_{20|0.015}$= $111.93
100.00+(2-1.5000)$a_{30|0.015}$= $112.01
Then he says The lowest price $111.93 occurs for n=20, i.e for redemption 10 years after issue.
Now finally for my question: Why is the lowest price of these three the correct answer? Why is $111.93 the highest price which an investor can pay and still be certain of a yield of 3%?