M&M questions with yield and coupon rate of debentures given

I’ve seen a few questions which give the issue of debentures and both the coupon rate and the yield. For example (These are my figures. I’ve not copied the question identically from one that’s published)

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Un-geared company $300
Cost of equity 10%
Tax rate 25%

Issue debentures $100 million
Coupon rate = 4% Yield = 5%

Calculate the cost of equity after the debentures are issued.
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I calculated the market value of debt based on the yield, the same as I would in a normal WACC calculation because the value of the interest paid doesn’t change (Market value of debt x yield = Book value of debt x coupon rate)

Market value of debt = 100 / 5% x 4% = 80
Market value of equity = 300 + 80 x 25% – 80 = 240

Putting this into the equation for Cost of equity I got

10% + (10% – 5%) x (80 x (1-25%) ) / 240 = 11.25%

This was wrong. According to the solution given I should have used the book value of debentures with the yield.

10% + (10% – 5%) x (100 x (1-25%) ) / 225 = 11.67%

It doesn’t make sense to me to use the yield with the book value. My understanding is that the yield given is almost meaningless at issue time. The yield is based on the actual interest paid / market value so unless the market value has changed the yield will be the coupon rate. If the coupon is less than the “required yield” at the time of issue there is an argument that the debentures wouldn’t sell initially at that price anyway. No-one buying a debenture at 100 would be happy that the yield had increased to 5% because that would mean the value of their capital investment had dropped. We normally have questions with a yield that is different to the coupon rate because they were issued some time ago and conditions have changed now and they are being traded which has resulted in the value dropping.

I’m obviously confused by this. I thought I understood M&M questions but having the yield brought into the question has thrown me. Please will you help me understand where I’m going wrong.