01 Volatility Managed Portfolios PDF

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Summarizing what I learn from the assigned research paper and then discuss this paper in the class through oral presentation....

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Presentation_1 Handout Class: Investment Management Topic: Volatility-Managed Portfolios (ALAN MOREIRA and TYLER MUIR, 2017) P1. / Research issue Managed portfolios: - Take less risk when volatility is high. Produce large alphas. Increase Sharpe ratios, and produce large utility gains for meanvariance investors (behavior of just the factors: the market, value, momentum, profitability, return on equity, investment, and betting-against-beta factors, as well as the currency carry trade: large empirical literature -> factors used summarizing pricing information contained in a wide set of assets). - Volatility timing: increases Sharpe ratios (changes in volatility are not offset by proportional changes in expected returns). - The strategy is contrary to conventional wisdom it takes relatively less risk in recessions). - The strategy rules out typical risk-based explanations and is a challenge to structural models of time-varying expected returns.

P2. / Methodology used to address research issue. - Portfolio Formation: + (ft+1: buy-and-hold portfolio excess return + ˆ σt2( f ): a proxy for the portfolio’s conditional variance (previous month’s realized variance)

+ constant c: controls the average exposure of the strategy. (use the full sample to compute c: managed portfolio has the same unconditional standard deviation as the buy and-hold portfolio). - The optimal portfolio weight:

(.)

- Empirical Methodology: + a positive alpha: volatility managed (~ volatility timing) strategy increases Sharpe ratios relative to the original factors (~ expands the mean-variance frontier)

P3. / Detailed of the data used in the study Daily (on exchange rate changes) and monthly (on returns) data: - Kenneth French’s website: {excess market return (Mkt), size factor (SMB), value factor (HML)} FF3 (1996), momentum factor (Mom), {profitability factor (RMW), and investment factor (CMA)} FF5 (2015). - Hou, Xue, and Zhang (2014): investment factor, IA, and a return on equity factor, ROE. - Frazzini and Pedersen (2014): betting-against-beta (BAB) factor. - Lustig, Roussanov, and Verdelhan (2011): “Carry” or “FX” factor. Currency returns (Monthly high minus low carry factor formed on the interest rate differential, or forward discount, of various currencies). Construct the volatility measure (monthly data on returns + daily data on exchange rate changes for the high and low portfolios).

P4. / Main results - Volatility-managed portfolios offer large risk-adjusted returns and are easy to implement in real time (Sharp ratios are improved: does not strongly forecast future returns) - Run contrary to conventional wisdom (taking relatively less risk in recessions and crises yet still earns high average returns) - Utility gains from volatility timing for mean-variance investors of around 65% (much larger than those from timing expected

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returns). Sharpe new ratio:

, percentage utility gain:

- strategy performance is informative about the dynamics of effective risk-aversion (a key object for theories of time-varying risk premia). Single-Factor Portfolios: (Table 1, & Figure 3) - positive, statistically significant intercepts (α’s) in most cases (directly relate to a long literature on market timing, avoid large momentum crashes by timing momentum volatility) - Managed factor excess Sharpe ratio (or “appraisal ratio”) given by α/σε: the extent to which dynamic trading expands the slope of the mean-variance efficient (MVE) frontier spanned by the original factors). - The strategy loses money relative to the buy-and-hold strategy (e.g. 1960s: volatility is low: takes relatively more risk) - In contrast, large market losses: volatility is high (e.g., the Great Depression or recent financial crisis) -> strategy avoids these episodes. Volatility-Managed Factor Alphas:

Multifactor Portfolios: (Table 3, & Figure 2&3) - choose weights so that multifactor portfolio is MVE (multifactor MVE) for the set of factors. - MVE alpha is the right measure of expansion in the mean-variance frontier: a positive MVE alpha implies that volatility managed strategy increases Sharpe ratios relative to that of the best buy-and-hold. - MVE portfolio:

(Ft+1: a vector of factor returns; b: the static weights that produce the maximum in-sample

Sharpe ratio) - Construct:

(c is a constant that normalizes the variance of the volatility managed MVE portfolio to be

the same as the MVE portfolio).

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- if volatility were constant over a particular period, strategy would be identical to the buy-and-hold strategy and alphas would be Zero. Business Cycle Risk (Table 3, Figure 3: : volatility-managed factor has a lower standard deviation through recession episodes): - across the board for all factors, strategies take less risk during recessions and thus have lower betas during recessions

Transaction Costs (Table 4): - strategies survive transaction costs (even in high volatility episodes where such costs likely rise; the annualized alpha of the volatility-managed strategy decreases somewhat for the market portfolio, but is still very large)

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Leverage Constraints (Table 5): - Typical investor can benefit from strategy even under a tight leverage constraint. - Constraint on leverage: weight: must be less than or equal to 1 (no leverage) or 1.5 (standard margin constraint) - Volatility timing weight:

; static buy-and-hold weight:

)

- Compute the weights and evaluate utility gains: risk aversion γ + high risk-aversion: this constraint is essentially never binding and their utility gains are unaffected + risk neutral will desire infinite risk exposure, and hence will do zero volatility timing (wt will always be above the constraint) + risk-aversion is low enough: some way to achieve a large risk exposure when volatility is very low (Concern: very high leverage might be costly or unfeasible -> implement strategy using options in the S&P 500, which provide embedded leverage. There may be many other ways to achieve β > 1—options simply provide one example) - Whenever the strategy prescribes leverage, use the option portfolios to achieve desired risk exposure.

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Volatility Comovement: - whether one can implement our results using a common volatility factor? (answer: does not drastically change the results) - compute the first principal component of realized variance across all factors and normalize each factor by 1/RVtPC (contrast to normalizing by each factor’s own realized variance)

- For most factors, the common volatility timing works about the same except the currency carry trade. Horizon Effects (even at lower frequencies, there is a negative relation between variance and the price of risk): - rebalancing it once a month and running time-series regressions at the monthly frequency (whether our results hold at lower frequencies?) - Less frequent: perspective of macro-finance models: at quarterly or annual frequencies: understand the full dynamic relationship between volatility shocks, expected returns, and the price of risk - studying the dynamics of risk and return through a VAR (a convenient tool to document how volatility and expected returns respond dynamically to a volatility shock over time) - run a VAR at the monthly frequency with one lag of (log) realized variance, realized returns, and the price-to-earning ratio (CAPE from Robert Shiller’s website)

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+ expected variance: spikes on impact, but shock dies out fair quickly, consistent w/variance being strongly mean-reverting + Expected returns: rise much less on impact but stay elevated for a longer period. - form portfolios (as before: weights proportional to monthly realized variance), but now we hold the position for T months before rebalancing, then run our time-series alpha test at the same frequency: +

, ft→t+T: cumulative factor excess returns from buying at the end of month T

- Alphas are statistically significant for longer holding periods but gradually decline in magnitude (alphas decline more slowly than the VAR suggests; these results are broadly consistent with the VAR in that alphas decrease with horizon)

General Equilibrium Implications (Macrofinance Models) - findings pose a challenge to macro-finance models - many equilibrium asset pricing models have largely ignored the risk-return trade-off literature, which runs regressions of future returns on volatility, because the results of that literature are ambiguous and statistically weak

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- Assess the statistical power of the approach by studying the predictions of four leading equilibrium asset pricing models (calibrate each model according to the original papers and simulate stock market return data for a sample of equal length to the historical sample): no model comes close to reproducing the findings in terms of alphas or appraisal ratios

- These results: the volatility-managed portfolios pose a fresh challenge to these models - The 4 leading models generally feature a weakly positive covariance between effective risk aversion and variance (typically have risk aversion either increasing or staying constant in bad economic times when volatility is also high). - The positive alpha -> empirically imply: this covariance is negative

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