Extraction of Inflation Expectation from bond yields In my introductory monetary economics course we have learned that inflation expectations can be calculated by comparing yields on index-linked and conventional bonds of equal maturity and assuming a risk neutral investor. One simple example given is:



A risk-neutral investor is indifferent between 1-year conventional and index-linked gilts (issued at par value, i.e. p = F) if expected returns until maturity equal:
$$ (1+i) F = (1+r) (1+\pi) F $$ so knowing $i$ and $r$ (from financial data), we can calculate $\pi$, the inflation expectation. 



Now this I am completely fine with, but then the lecture goes on to the following example:


  Extracting inflation expectations from gilts (assuming risk
  neutrality) Use Tr 4.5pc ‘42: i = 1.71%, and IL 0.625pc ‘42: r =
  −1.66% [Tradeweb] $$(1 + 0.0171)^N = (1 − 0.0166)^N (1+\pi^e_N)^N$$ 
  giving $\pi_N^e \approx 3.43%$ break-even inflation for a horizon of N =
  23 Years


My question is: how can this calculation be valid when the coupon rates are different for the two gilts? What are the assumptions (eg issued at face value)? 

My thinking is that to extract inflation expectations, we must equate the expected returns of the bonds until maturity, just as in the earlier simple example. The rate of return is however clearly tied to the coupon payments, which each need to be multiplied at the relevant accrued yield, leading to a much more complicated equation for the inflation expectation.
Yes, the number you give is an approximation. If you wanted to calculate the true economic breakeven, you would need to include the effect of the lag used in the indexation process (which includes known inflation information), coupon cash flows, seasonality of inflation, etc. If there is a maturity mismatch, then you would need to decide how to handle that as well.

However, the deviation between the economic breakeven and the approximation is normally small. In fact, the market convention is (or at least it was) to ignore compounding and just use the spread between quoted rates.

You would need a full fixed income pricing suite to calculate the true economic breakeven, so a course would stick to the approximation.