Investment Portfolio Management Final Notes - Last two Units
Investment Portfolio Management Final Notes - Last two Units 421
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This 12 page Study Guide was uploaded by Austin Siemion on Thursday April 14, 2016. The Study Guide belongs to 421 at University of Miami taught by Sandro Andrade in Spring 2016. Since its upload, it has received 20 views.
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Date Created: 04/14/16
…Notes before this in Slides 4 There are Issues with Statified Sampling: (1) sensitiveness to arbitrary definitions of strata – why is large cap threshold $10b/why only two cap bins/which sector does Gen Electric really belong to (2) Inflexibility and lack of optimality in trading-off E[alpha] vs. Active Risk – some consumer staples not particularly well-ranked by UM super combo o We need a more comprehensive framework to QUANTIFY AND TRADE-OFF Active Risk vs. Expected Active Return! Our goal: Choose portfolio with sufficiently large Expected Active Return, that doesn’t take too much Active Risk o They depend on a vector E[ ] and a matrix V o Vectors and matrices greatly simplify the algebra, and our quantitative signals determine E o V can be estimated from past stock returns o Having both E and V, and a target for Active Risk, we can find the optimal portfolio rPis the future return of the portfolio, B is the future return of the benchmark, and these equations define E and Active Risk: NOTICE! Now the focus is not on absolute return, but return relative to a benchmark, also known as Active Return or alpha Portfolio Expected Alpha and Portfolio Expected Active Return mean the same thing Considering again the Rolling Backtest of the UM Super Combo Top 100 equally-weighted stock portfolio, Portfolio Active Risk = 16% per year, much larger than the Active Risk of LSV and Arrowstreet funds! We have a Universe of N stocks, and a benchmark against which our clients assess our performance and alphabetically ordered by ticker symbol Side note – there are less than 3000 stocks in the Russell index because it is re-constituted at the end of each year, after some stocks drop because of M&A, going private, and bankruptcy etc. Define Active Return (Alpha) of each stock as: o So, now there is a 1xN vector of active returns o E represents your best forecast for the active return of a stock V is the NxN matrix of stock return variance/covariances, which can also be written using volatilities (standard deviations) and correlations A portfolio of stocks is denoted 1xN vector of portfolio weights, w o w < 0 represents a short position in stock i o Define 1 = [1 1 1 1 1] (1xN vector of 1’s) o We focus on fully-invested portfolios because this class is about security selection, not market timing o We care about two fully invested portfolios, W , oPr portfolio, and W ,Bthe benchmark portfolio Note, there is no tilde over the W because the portfolio weights are not uncertain, benchmark weights are known to us and portfolio weights are chosen by us Remember: Matrix multiplication AB is defined if and only if the number of columns of A is equal to the number of rows of B Remember: Matrix transposition (the ` symbol) shifts columns and rows The future returns of our portfolio and of the benchmark can be calculated by the weights of the portfolio vector x the future returns vector, which can then be solved to obtain Portfolio Active Return: The Expected Active Return of a portfolio is the best forecast of the return difference between that of the portfolio and the benchmark. It is the weighted-average of the expected alphas of its constituent stocks, where weights are portfolio weights Active Risk of our portfolio is the standard deviation of the active return which translates in the square root of portfolio weight – benchmark weight paired in a Matrix with portfolio weight – benchmark weight transposed The Active Risk (standard deviation of return difference) of a portfolio depends on how portfolio weights deviate from benchmark weights, and on the structure of volatilities and correlations of all stocks o Portfolio Active Risk increases when deviations from the benchmark occur in particularly volatile stocks (high standard deviation, ) o It is mitigated when a given deviation from the benchmark is partially offset by a counterbalancing position in a positively correlated stock (overweight GM, but underweight Ford relative to the benchmark) We used matrix notation because if we were to use the extensive, non- matrix formula, it would not work when N is a large number! (say, N=3000) Excel has special functions to do matrix transposition, multiplication, and inversion o Enter formulas using CTRL+SHIFT+ENTER o If the outcome of a matrix formula is another matrix instead of a scalar, and if you want to see the entire outcome, precisely select the destination range and then re-enter formula using CTRL+SHIFT+ENTER MMULT formula only works for two arguments so we must nest two MMULT formulas to compute Portfolio Active Risk: Calculate the Expected Active Return and Active Risk of any given portfolio using the formulas: For any target level of Active Risk, and given portfolio constraints, the optimal portfolio W hPs the highest Expected Active Return The critical difference to note once again is that here, return and risk are measured relative to a benchmark! Using calculus, the optimal unconstrained fully-invested portfolio shows that deviation from the benchmark is a pure long-short portfolio and that the nature of it does not depend on target Active Risk Y o The weights of the portfolio are simply scaled up or down depending on target Active Risk Y (managers with high Y should simply scale up the same deviations from the benchmark as managers with low Y In small scale portfolio optimization, given the benchmark, we can’t choose any E[alpha] o Logical constraint is involved; for internal consistency, the Expected Active Return of the benchmark has to be equal to 0 such that W E[alpha]`=0 because the benchmark cannot beat B or lag itself! o Additional constraints on the portfolio choice might be that the weight for all stocks is Wp>0 which means no-shorting Information ratio does not depend on Y in an optimal unconstrained portfolio, so it does not change as Active Risk is changed, however, in a no-shorting or constrained portfolio, with higher Active Risk target, portfolio constraints such as no-shorting become more costly in terms of reducing performance (Expected Active Return/Active Risk moves one for one until constraints are involved) Portfolio math shows us how to QUANTIFY and TRADE-OFF Active Risk and Expected Active Return o **But in practice, we can obtain E[alpha] from current stock ranks, using Grinold-Kahn approach and V from past stock returns using a Factor Model** Grinold-Kahn – greatly simplifies implementation procedure, need not forecast alphas with laser-like precision because accuracy is apt to be fairly low, keep process simple and moving in the correct direction Step 1 – Use ranking system based on Q-signals, order all stocks in Universe from most attractive to least attractive o Extract data, and vlookup(ticker,$search$area,reference rank, FALSE) to rank from N1 o Step 2 – Calculate Trimmed Percentile Ranking (TRP) for each stock to trim out top .6% and the bottom .6% (to avoid outliers) Step 3 – Calculate the non-neutralized (NN) expected active return for each stock given its TPR using formula NORM.INV(TPR,0,2%) which range from -5% to +5% normally distributed with a mean 0% and a standard deviation 2% per year Step 4 – Given Non-neutralized alpha and benchmark portfolio weights, calculate the expected active return for each stock o The last step forces Expected Active Return of benchmark to be equal to 0, which is necessary for internal consistency o It also shifts Expected Active Return while maintaining shape and standard deviation Conservative, as expected alphas are scaled down relative to backtest alphas, which is wise to do for two reasons: o Competition – other investors learn of signals and invest based on them, and this trading behavior bids up price of undervalued and depresses price of overvalued (narrows mispricing) o Overfitting – backtests are too optimistic because we played around with different definitions of signals and chose those delivering best performance in the sample, which doesn’t guarantee that the chosen definitions will deliver best performance out of the sample (in-sample performance improvement is not repeatable) If we were not to trim sample, then results would show that small rank differences led to large differences in expected alpha (BAD) We avoid large alpha differences for small rank differences, scale down prospective alphas, and enforce logical consistency, on a scale that is reasonable along other dimensions (normally distributed) Factor Model – unstructured, structured, hybrid Use historical excess returns for all stocks in the Universe – assuming that, on average, volatilities and correlations are sufficiently stable over time Unstructured: Use the sample V without imposing any constraints on its structure o Step 1 – choose sample period and sample frequency (152 weekly excess returns over last 3 years) o Step 2 – for each stock, calculate sample standard deviation and covariance o Step 3 – arrange in matrix to get V=V unstructured o Annualize by multiplying weekly by 52 o PROBLEMS: Due to very high estimated parameters and data points, there are not enough degrees of freedom! V, though correct on average, will contain some very poor estimates – among millions of correlations, some of them will be, just by chance, unusually very high or very low, that are very unlikely to repeat in the future If the goal is to find optimal portfolios, then the poor estimates create a serious problem – the optimization will effectively “zoom in” in the poor estimates, leading to nonsensical optimal portfolios with extreme long- short positions Say, sample correlation between two stocks (with expected alphas of 1% and -1%) is unusually high at 0.99, much higher than the correlation of future returns between the two stocks – the optimal portfolio will show ridiculous high long positions backed up by ridiculously high short positions We can use unstructured method to measure Active Risk because it’s correct on average, but there is a problem when unstructured is coupled with optimization Structured: greatly reduces number of parameters to be estimated by pre-imposing some structure on V (by assuming all co-movement between stocks is due to shared exposure to factors identified by analysts in advanced) o Changes the way correlations are estimated, NOT affecting volatilities o “Idiosyncratic” Return - the component of the stock that is unrelated with Market returns, also the idiosyncratic returns between two stocks are unrelated among themselves, cov(stocks) = 0, cov(stock+mkt) = 0 o We must decompose the variance of the stock to differentiate systematic from idiosyncratic variance, knocking out the covariance of both as they are uncorrelated …Notes after this in slides 4 Slides 5 –Cost Control Cost control matters – average all-in trading cost per trade, tax rate for short-term capital gains Cost control happens at two stages o Trade execution – move from existing to target portfolio at the lowest possible trading cost o Portfolio construction: integrate transaction costs and taxes into Portfolio Optimization Important, particularly for managers of large portfolios to control costs so that potential investment profits are not dissipated in excessive or inefficient trading Portfolio construction – cost-aware choice of quantitative signals (avoid those that require large turnover, analyst sentiment combo is costly to trade), transaction cost model accounts for costs moving from existing to optimal target portfolio Trade Execution – optimal target portfolio is passed along from portfolio managers to traders who attempt to transact it at minimal cost, frequently relying on execution algorithms Trading costs have explicit and implicit components o Explicit – commissions and fees o Implicit (higher weight) – you don’t pay for them separately, they show up on the prices you trade at (bid-ask spread, market impact, slippage) Average trading costs per trade – 45 bps (0.45%) Trading costs vary across stocks – in particular, trading costs increase as market capitalization decreases Accounting for trading costs matters a lot in Quantitative Investing! Beware of paper profits! o Ignoring transaction costs of .45% per trade (slippage=0.45), top bucket had 21% per year, but when we include costs, the top bucket has -2% per year due to very high turnover Commissions and fees – paid to brokerages, exchanges, and regulators for services they provide o Access to other market participants o Operational infrastructure (handling post-trade activities) clearing (regulatory reporting and monitoring) settlement (delivery of securities in exchange for payment) o Improved security of transacting o In short positions, fees are also paid to stock lenders Bid – ask spread – difference between the best current limit order to buy and the best limit order to sell is the current bid-ask spread Bid – ask spread and trade orders o Market order: liquidity demanding, pays (one half of) the bid-ask spread, executed with certainty o Limit order: liquidity providing, earns (one half of) the bid-ask spread, may not be executed Bid – ask spread determinants o Processing costs – to maintain the trading platform infrastructure, tiny nowadays o Liquidity providers’ Inventory Risk: after transacting with you, and before he covers his position with a subsequent trade, price moves in adverse direction to him (bid-ask decrease with trading volume and increase with volatility) o Liquidity providers’ Asymmetric Information Risk: on average lose when they trade with better informed counterparties (bid- ask increase with greater chance that important info is known to only small subset of people o Competition among liquidity providers o Regulatory constraints (minimum tick size) Market Impact o Extra cost for trading more than just a small number of stocks – larger trades get executed at worse prices than what current best bid or best ask indicate o For institutional traders, this is the most important trading costs, and they must break down large orders into small pieces that are executed throughout a certain window of time Liquidity Fragmentation – stock liquidity in the US in more than 10 separate limit order books for a typical stock, each held at a different trading platform Regulation NMS (National Market System) – attempts to ensure that certain orders are executed at the best price available across all limit order books (if better price is available elsewhere, the trading platform wont execute the order) o There are exceptions – Latencies and Intermarket Sweep Orders Dark pools – crossing networks, participants send trading intentions into dark pool, anonymously matched and trades executed o Un-matched trades due to supply/demand imbalances are not made public o External price: trades executed at mid-point btw best bid and ask in pre-specified exchange (large orders inside pool will not move prices!) o Dozens of dark pools, managed independently or within large brokerages o Designed to minimize market impact for large investors Slippage o Opportunity cost of delayed/missed trades – flip side of being patient to minimize market impact because prices could move against you before you actually trade the entire amount you planned to o Not just risk, but “cost” because it negatively impacts your performance o If you are right about forecasting a price movement, on average, then slippage will negatively affect you on average o Trade – off: (Bid-ask + Market Impact) vs. Slippage Aggressive/Impatient/Liquidity demanding trading – higher market impact and bid-ask costs, but smaller slippage o Historical perspective in US exchanges – smaller bid-ask spreads… but less depth (higher market impact good for smaller investors who don’t suffer as much from marker impact) Costs 2 – trade execution – goal is to execute the trades leading to target optimal portfolio as completely and cheaply as possible o Order scheduling: How to break down orders and distribute over time? o Order choice: Which kind of orders? – market order (aggressive) or limit order away from best bid/ask (least aggressive), some ECNs allow hidden limit orders with less priority o Order routing: Where to send orders? In which sequence? – more than 10 limit order books, dozens of dark pools ..few more notes on slides 5
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