I agree with

@Dr. Jayanthi Sankaran 's approaches. I guess we have two versions of the problem: one solving for the price of a 4.5% coupon, one solving for the price of a 7.0% coupon. The unmentioned problem is that the question has a

**design flaw**: it violates the "law of one price" which requires the one-year discount factor to be the same for all three bonds. Specifically, let's say the 5 bond is accurately priced; if so, the one year discount factor, df(1.0), is given by $97.50/105.00 = 0.92857. In this case, the price of the 8 coupon bond must be 0.92857 * $108.00 = $100.286. Then we can solve for either 7 or 4 /12 bond in the two scenarios below which are "internally consistent":

- $5.00 coupon bond priced at $97.50
- $7.00 coupon bond priced at
**???**
- $8.00 coupon bond priced at $100.286

Or for that matter we can solve for:

- $5.00 coupon bond priced at $97.50
- $4.50 coupon bond priced at
**???**
- $8.00 coupon bond priced at $100.286

We can still solve per the intended method. For example, in the case of the 4 1/2 coupon, X = position in 5 coupon bond = 116.67% (see Dr Jayanthi's above), such that price of 4 1/2 bond = $97.036. The law of one price is here satisfied because 97.036/104.50 = 0.92857; ie, same discount factor. That's why this flaw was identified: you should have a quicker method to getting the price, per the shared discount factor, with 0.92857*104.50. But notice also that given this simple scenario

**we can also just use a simple ratio** given by 104.50/105.00 * 97.500 = $97.036 (both bonds are not really required). This question type is not well designed. BTW, GARP is aware, we reported this carefully at the time (2 years ago) when a customer pointed it out. I hope that helps!

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