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This conditional CAPM uses monthly returns from January 2022 through December 2024. The optimal portfolio returns come from the weights in efficient frontier (long-only and short-allowed), the market proxy is the QQQ ETF, and excess returns are measured relative to a zero risk‑free rate. The state variables are the prior-month market return and the lagged 12-month rolling market volatility (both standardized). Interactions between the market factor and these state variables allow the market beta to vary with business-cycle and risk conditions.

For the long-only portfolio the market beta is about 0.90 (t = 10.18) and a small positive monthly alpha of 0.009 (t = 2.33) remains after conditioning. The interaction with lagged market return is mildly positive (0.16, t = 1.95), hinting that beta rises after strong markets, while the interaction with lagged volatility is negative but not significant. The negative coefficient on lagged market return alone (-0.015, t = -2.54) suggests some mean reversion beyond the beta’s own variation. The model explains roughly 82.5% of the return variation, so conditional CAPM structure fits this portfolio well.

For the short-allowed portfolio the unconditional beta is indistinguishable from zero (-0.02, t = -0.04), but the interaction terms are large and significantly negative (-1.07 on lagged return and -1.45 on lagged volatility), implying the portfolio’s market exposure shrinks when the prior market is strong or volatility has been elevated. A sizable positive alpha of 0.059 (t = 2.05) appears, yet overall fit is weaker (R² ≈ 0.29), so much of the return variation is not captured by these conditional betas.

Overall, the long-only portfolio behaves like a high-beta allocation with modest time variation in beta, while the short-allowed portfolio relies on timing effects that reduce market exposure in higher-risk or momentum phases.