Factors

(a) Prove Proposition 6A.20. Then compute the monthly BAB return when $\beta_L = 0.7$, $\beta_H = 1.3$, $r_f$ = 0.5%, $r_L$ = 1.2% and $r_H$ = 1.5%. (b) For i.i.d. monthly returns, show that the $t$-statistic of the mean is approximately the annualised Sharpe ratio times $\sqrt{\text{years}}$. Check it against the WML row of the IIMA table in 6A.12, and compute how many years the market factor needs for $t = 2$ at its Sharpe ratio. (c) SMB has a negative mean over 1994–2025 in India. List three explanations and say how you would distinguish between them. (d) Explain why a long-only "Nifty200 Value 30" index is not the HML factor, and what regression you would run to measure its value exposure.