How to find average book value?

The average book value takes the start and end book values because that are the transactional points, i.e. when you invested and when you divested.

Just like purchasing a stock, your return depends on the value of the stock when you purchase it to when you sell it. What the value is in between these transactions can be interesting, but irrelevant to your return.

Net profit is totally different – net profit for all years is relevant to your return calculation, because this is the return you get on your investment and available to common shares via e.g. dividends every year.

Hope that helps!

Average accounting return is average net profit over average book value.

Average book value is taken by the start and end of the period divided by 2 because it assumes the book value trends from the start value to the end value in a straight line.

So expanding on your example, say across 4 years, profit and book value are:

Return on equity:

$$\begin{gathered} =\frac{{\displaystyle (15,000+25,000+20,000+10,000)/4}}{{\displaystyle (50,000+15,000)/2}} \\ =\frac{17,500}{32,500} \\ =54\% \end{gathered}$$