Investment Management Cheat Sheet, Cheat Sheet of Investment Management and Portfolio Theory

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FIN367 – Michael Eng Ch1 – Investments Environment -Real assets vs financial assets – real = capacity to create & illiquid, finance = claims on value of real assets & liquid -decide as to how much to save, how much to allocate across asset classes, specific securities in each class -Top-down (allocate portfolio by asset class then specific securities) vs. bottom-up (select specific, attractive securities then diversify across asset classes) -passive (very diverse portfolio) vs. active management (timing market and finding mispriced securities) -prices set in equilibrium – if price is low, investors will buy and increase price; if high, investors sell and price drops -risk-return tradeoff – iff take risk, get extra returns -market efficiency – prices reflect available info to everyone so it’s hard to beat market Ch5 – Risk, Return, and History Gross Return = simple return = gross – 1 Returns are measured over any holding period Frequently normalize returns to annual rate: where period return = r, period in years = T, number of periods/year = n = 1/T APR = nr –> APR is less than effective. annualized rate (EAR), more so as n decreases (monthly->weekly->daily) -two ways to analyze returns, real(rate adjusted for inflation) and nominal (rate for US$) or nominal = real + inflation -How is real rate set: Equilibrium at which supply of capital (savings) = demand for capital (investments) -As R increases, supply (savings) increases, demand (investments) decreases (higher hurdle rate = smaller NPV) -Government policy (debt, monetary) also affects Ri -only in past 50 yrs, inflation has been close to expectation -returns are uncertain – have to use probabilities to estimate returns and almost always have variance -using historical data assumes returns have a stable distribution over time -limit 1 – only have limited observations (80 years) -limit 2 – return dist. may change over time -instead of estimating whole return, estimate excess return: Excess return(top), risk prem. (bottom) -better b/c excess return dist. probably more stable w/ time; removes risk-free rate which is known to change w/ time; excess returns insensitive to predictions of inflation; excess returns are what investors care about since risk free assets are the alternative investment Risk/reward: -Average returns: geometric and arithmetic means: A ≈ G + ½(variance) -which to use? – G is good for past performance (shows annual return actually realized, = IRR and CAGR); A is expec. return if held for a single period & good for forecasting -VaR – value at risk – loss corresponding to very low percentile of entire return distribution (how much could you potentially lose in a worst-case scenario) -Skewness – neg. skew implies higher chance of neg. events -Kurtosis – measure of how fat tails are Skew = kurtosis = -often assume return dist. are normal because: Reasonable estimate – empirical dist. of returns close to norm. (although fatter tails and neg. skew); CLT – sum of small random variables (continuous returns) usually norm. Easy to work with – norm dist. described well by mean & standard dev.; portfolio of normal returns also normal (helps with portfolio analysis Weights: Portfolio returns (can use expected values): -risk is characterized by variance and std. dev. -need to take variance of each assets returns and the covariance between each pair of asset -need covariance to compensate for correlated data between the two assets (results in cancelling out) – covariance important because it is most indicative of diversification benefits Correlation – easier to understand Cov. -btwn 1 and -1, 0 = no correlation; risk-free asset returns are always uncorrelated with any other asset returns -equity returns historically quite positive and averaged excess return of 7.5% (st. dev. 20.46) -did have periods where they performed poorly (30s and 2000s) -high real returns compared to rest of world Ch6 – Capital Allocation -risk averse – avoid even fair games – degree of which is how much you will pay to avoid -risk neutral – don’t care either way -risk loving – will seek out fair games -visualized by diminishing marginal utility of wealth -certainty equivalence – investor indifferent between risky payoff and this certain payoff -for portfolio preference – investors are assumed to be risk averse – want highest return for given risk and lowest risk for given return – higher return and lower risk portfolio are seen to dominate other portfolio Utility function: U = certainty equivalent; A = risk aversion -assumes investor has mean-variance pref. and only cares about those metrics, reasonable if normal returns Indifference curves: -portfolio allocation – decide risky vs. risk-free and which risky assets -in a portfolio/asset (a) and risk free asset (f) allocation, r = rf + (wa)(ra-rf) and  = p x weight in portfolio/asset (b/c cov(rp, rf) = 0) -capital allocation line (CAL) – investment opportunities generated by a risky portfolio/asset and risk-free asset; y = returns, x = , slope = sharpe ratio of risky asset, intercept = rf rate -borrowing rate>risk free rate so CAL line with weight in risky asset>1 is usually less steep -CML = CAL where risky portfolio = market portfolio – passive investor chooses optimal weight for portfolio along the CML – this tactic outperforms active management; if market efficient, difficult to outperform it Optimal portfolio for investor with aversion A: -allocate more to risky asset when premium higher, var. smaller or investor less risk-averse Diversification -decrease portfolio variance through diversification and reduction of idiosyncratic risk across the portfolio -cov. and correlation measure how things tend to move together -diversification with n assets gives variance: -as n increases variance decreases; as n increases, variance approaches cov,(shows limits to diversification); a diversified portfolios variance is driven by average covariance of assets, not variance -risk is driven by covariance not variance -can diversify away idiosyncratic, not systematic/market, risk -global diversification removes country-specific market risk (in US reduces  by 8% from 22% to 14%) Ch7 – Optimal Risky Portfolio For a set of risky assets E and B: -Min. Variance Portfolio – point at which variance between two assets is minimized – all portfolio combinations below this point are dominated by the points at the MVP & above -tangency portfolio – weighted portfolio btwn two assets that maximizes the Sharpe Ratio – optimal portfolio -CAL through tangency portfolio dominates all others -separation property – all investors will choose the same portfolio of stocks and bonds and will adjust that fixed portfolios weight in relation to the risk-free asset’s weight -can theoretically take the opportunity sets between various each pair of possible assets out of as set of assets and create a curve from the result of all the variances: -Minimum variance frontier – portfolio with lowest possible variance for a given return -can add constraints when calculating the frontier and then compare the frontier to a risk-free asset and CALs -steepest CAL has portfolios that dominates all others -slope of CAL = Sharpe ratio of risky portfolio -optimal risky portfolio is the CAL lying tangent to the efficient frontier – tangency portfolio -choose point on that CAL depending on risk aversion -everyone will have the same CAL but choose diff. points on the CAL to maximize their Utility 4 steps to portfolio optimization: 1.estimate means and covariances of investment options 2.calculate min. variance and efficient frontiers 3.find portfolio with highest Sharpe ratio 4.calculate optimal CAL by connecting rf asset w/ tangent portfolio – choose portfolio along this line Seperation property – can choose optimal portfolio by: first choose which risky assets to invest in; then how much to invest in risky portfolio vs. risk free -basis for mutual funds – optimal risky portfolios vary b/c of: tax considerations; social responsibility; liquidity needs; etc. -theories apply to both asset allocation and security selection but they are usually done separately – limits on how many securities you can compare and estimates of means more reliable at portfolio/class level