Dividend adjustment conventions

(a) Prove (1.11) and show that the difference between the multiplicatively adjusted return and the simple total return on the ex-date is exactly $x\,d/P_c$, where $x$ is the adjusted return. (b) An alternative "additive" convention subtracts $d$ from every price before the ex-date. Show that it distorts returns on all earlier dates, give the formula for the distorted return, and show by example that it can produce negative prices for a stock with a long dividend history. (c) Which convention preserves log-return volatility estimates over long histories, and why?