When redemption is at the option of the borrower, then you should consider the worst case from the point of view of the investor (as it's the investor who decides how much they are willing to pay). The worst case for the investor is a capital loss being received at the earliest date, and the capital gain being received at the latest date.
It's questionable what is the right thing to do in the scenario when redemption is at the option of the investor. Note the wording of syllabus objective (xi)2 for the UK exams (I think the IAI syllabus is the same):
"Calculate upper and lower bounds for the present value of a fixed interest security that is redeemable on a single date within a given range at the option of the borrower"
So, it appears to me that this question is beyond the CT1 syllabus, and I'm not completely convinced by the answer given. Here's what I think:
We can never be certain of getting a particular yield on an optional redemption date bond, as the yield earned depends on when redemption occurs, which we don’t know. In this question, if we assume the earliest redemption date (15 years), I get a price of 101.34. If we assume the latest redemption date (20 years), we get a price of 101.54. The price increases with duration, as there is a capital loss. You could calculate the price at intermediate durations – this should give steadily increasing values from 101.34 to 101.54.
The investor wants a yield of 8%. If they choose to pay 101.54, then if the bond is redeemed after 20 years, they get a return of exactly 8%. If the bond is redeemed earlier, then they get a return of less than 8%, which is a problem. So they cannot pay 101.54 and always get a yield of 8% - if they paid that price, there would be occasions where the yield is less than 8%.
If the investor pays 101.34, then if the bond is redeemed after 15 years, they get a return of exactly 8%. If the bond is redeemed later, then they get a return of more than 8%. And that’s fine. So 101.34 is the highest price they can pay to be certain of obtaining a yield of 8% on their investment. If they paid more, then we could find a redemption date on which they have a yield of less than 8%.
Based on this, I'd say the answer was 101.34 (ie the answer you'd get based on assuming the worst case for the investor).
The counterargument to this is that, as the investor has the option, they would never choose to redeem the bond at a date that was disadvantageous to them. So if they paid 101.54 (based on an 8% yield and redeeming after 20 years), then they would never choose to redeem early and so never receive less than an 8% yield.
So, as I said, it's questionable what's right here.
