Potfolio Management Flashcards
(56 cards)
long term corp bond have lower risk than LT govy but higher return
equity and stock do not follow normal distribution, negatively skewed=large downside deviations, Mean < Median < Mode, left tail; excess kurtosis (fatter tail)=large upside and downside deviations
expect frequent small gains and a few large losses
positive excess kurtosis
leptokurtic distributions, preferred vs playtokurtic cuz less risk;
higher peak, fatter tail, more outliers;
greater concentration of data points around the mean; greater chance of extreme positive or negative outcomes;
calculated by subtracting 3 from the kurtosis value, more peaked;
Kurtosis describes the “tailedness” or “peakedness;
A normal distribution has a kurtosis of 3
investors are risk adverse;
risk adverse investors would choose the portfolio in higher indifference curve;
investors do not minimize risk, but trade off;
indifference curve=investor have no preference at each point on curve, higher curve=more risk adverse investors, want more return at same risk; lower curve=less risk adverse investor
convex cuz its x axis is variance, not sd;
flatter curve, lower slope, more risk taking, dont need that much return to take risk;
where indiff curve touch CAL as tangent, optimal portfolio max return per unit of risk on CAL;
above CAL are not achieveable, below is inefficient
capital allocation line
is the risk/return line of a risk free asset+a risky asset; x axis is risk, y is return; x at 0 is rf asset moving towards risky asset point, somwhere in the middle is portfolio;
SA=σA
Var(Rp)=σA^2WA^2+σB^2WB^2+2WAWBCovAB;
CovAB=ρσA*σB=sigma[(X−EX)(Y−EY)]/(n-1)
when asset B is risk free asset (0 covariance and volatility):
Var(Rp)=σA^2WA^2
σ(Rp)=σAWA
CAL:
Rp=Rf+[(RA-Rf)/σA]*σ(Rp)
CAL from expected return formula
E(Rp)=WARA+WBRB
=WARA+(1-WA)RB
assume B is rf asset
=Rf+(RA-Rf)WA=risk free rateweight of risk premium
take σ(Rp)=σAWA from CAL
CAL=Rp=Rf+[(RA-Rf)/σA]σ(Rp)=>y=intercept+slope*x=straight line
slope is the sharpe ratio for portfolio A; compares return of port with risk;
investor will have a portfolio mix where CAL is a tangent to the indifference curve, a further indifference curve on same CAL is less risk adverse on a different set of curve (more risk more return)
covariance
how far the average distance from mean of asset A and B are together;
times the distance A*B, sum them, divide by n-1
r on calculator means correlation
SASBr=Cov(A,B);
correlation is the measure of linear relationship between asset A and B;
when correlation=1, portfolio’s standard deviation is the same as the weighted average of the individual assets’ standard deviations, meaning there’s no diversification benefit; σAB=WAσA+WBσB
plug in 1 in covariance equation: CovAB=(a+b)^2 so sd=a+b;
as corrlation decrease goes to negative, risk of the portfolio increase, less diversificatio benefit, expected return is unaffected by correlation
minimum variance frontier and efficient frontier
minimum variance frontier=given risk, returns and correlations, draw a cruve that shows the return at each level that has min risk; the whole curve; not all points are efficient;
efficient frontier=top half of curve; set of porfolios that gives max return for each level of risk (sd);
top half part of curve is most efficient (only invest here), area below that are are not efficient (cuz same risk lower return); see picture;
CAPM (capital asset pricing model)
CAPM determines the expected return a security based on only the systematic risk β assume unsysmtematic is eliminated; can also determine the required (fair) return based of its beta; required return and expected return are equal in equilibrium if points are on the SML line;
SML (Security Market Line) is linear respresentation of CAPM;
assumptions:
investors use mean-variance framework (risk/return);
unlimited lending and borrowing at Rf;
homogeneous expectations (market portfolio);
one-period time horizon (no compounding);
dividsible asset (can trade as much as we want);
frictionless market (no large cost);
no inflation and same interest rate;
equilibrium markets (fairly priced), investors price takers;
E(Ri)=Rf+βi(Rmkt-Rf)
beta is systematic risk (x), how much an asset’s price is expected to move in response to changes in the market, standized covariance of asset return with market return;
βi=COV i,mkt/σ mkt^2=correlation*(σi/σmkt)=>cor(i,m) * σ(i) / σ(m);
(Rmkt-Rf)=slope;
jensen alpha=forecasted return (actual)-required return(CAPM, whats fair)= determine if a portfolio is earning the proper return for its level of total risk assuming no other risk; positive=undervalue, above capm line; on the line, indifferent
graph: beta x axis, return y axis; if over the capm line, undervalued, buy;
when beta=1, expected return of portfolio gives the expected return of market;
CML vs CAPM: CML gives total risk of efficient portfolio, CAPM only gives fair return of systematic risk, appraisal tool, is your security fairly valued; slope if CML is sharpe, slope of CAPM is treynor; points above CML is unattainable under is inefficient; points above CAPM is undervalue, below is overvalued;
capm low beta might not meant low risk in total, unsystematic risk is not shown.
Applications:
preformance evaluation=analyze risk and return of active manager’s portfolio alpha;
attribution analysis=analyze the split of active manager’s portfolio alpha returns or beta benchmark portfolio’s return;
CML (capital market line)
with homogenuous expecations of risk, return and correlation, all investors have the same optimal risky portfolio;
for optimal portfolio (market portfolio, assuming market is efficient), we wont invest 100% on efficiency frontier but combine rf asset with efficiency frontier (minimum variance portfolio), this is tangential line which gives the steepest capital allocation line, the tangent point is on the efficiency frontier and have highest sharpe ratio (steepest slop so highest sharpe, (rA-rf)/σA);
not a equilibrium because equi plot all asset on the line SML but here only plots risky asset and rf asset.
CAL vs CML-
any asset has CAL: combine risky asset with rf asset;
CML=special case of CAL where we all agree on homogenuous risk, return and correlation, that passes through the theoretical market portfolio; any thing above CML does not exist, below is inefficient; optimal CAL=CML
CAL: Rp=Rf+[(RA-Rf)/σA]σ(Rp); asset A sharpe ratio
CML: Rp=Rf+[(Rm-Rf)/σm]σ(Rp) instead asset A, use most effcicient market port; market sharpe ratio
or Rp=Rf+[(Rm-Rf)*[σ(Rp)/σm]: total risk of portfolio determines the market risk premium we get;
if CAL slope>CML slope, security perform better than market on total risk adjusted abnormal return, which shouldnt exist assuming market is efficient; Abnormal return = Actual return – expected risk-adjusted return;
CML includes lending portfolios with positive allocations to the risk-free asset, the market portfolio with no allocation to the risk-free asset, and borrowing portfolios with negative allocations to the risk-free asset.
Since the line is straight, the math implies that the returns on any two portfolios on this line will be perfectly, positively correlated with each other. Note: When ρa,b = 1, then the equation for risk changes to sport = WAsA + WBsB, which is a straight line.
Asset character line: βi=Cov (i,m) / σ(m)^2)
CAPM: E(Ri)=Rf+βi(Rmkt-Rf);
if more than 100% allocation, borrowing to invest;
slope=sharpe ratio=excess return per unit of risk=(Rp-Rf)/σp;
sharpe ratio higher than market sharpe, superior portfolio;
systematic risk vs unsystematic risk
market risk=cause by macro factors, its measured by covariance of port returns with market returns; beta=sensitivity to market risk
unsystematic risk=firm specific and can be eliminated by diversification;
CAPM only systematic risk is rewarded with higher expected returns cuz it cant be diversified away, assume unsystematic risk is diversified under efficient market;
diminishing return where increase number of stocks to a point where total risk is at min (only market risk left)
one risk factor market model (returns generatign models)
the one factor is premium on market portfolio; e.g. capm at rf rate
Ri=αi+βiRm+ei
beta=sensitivity to market risk; beta (βi) measures the sensitivity of the rate of return on an asset (Ri) to the market rate of return (Rm).
alpha=unexplained persistent nonzero return;
ei=noise error should be cancel out; unsystematic return; variance of error=unsystematic variance of stock;
x axis market return y is port expected return
multi risk factor market model (returns generatign models)
if one risk factor (market) doesnt explain all non-diversified risk, use
E[Ri]-Rf=βi,1E[F1]+βi,2E[F2]+…βi,k*E[Fk];
Factors F are the expected values of each risk factor;
0βi,k are factor sensitivities for factors like macroeconomic factor, fundamental factor or statistical factor (no financial theory but just math trends);
e.g. Fama French 3 factor model:
risk factors are firm size, book to market ratio, excess return on market portfolio; small cap stocks outperformance large, high BV vs Market value (low price to book ratio, cheaper) outperform low BV vs MV ;
e.g. Carhart added 4th factor momentum, buy things going up cuz they continue to go up and earn risk premium on it; counter market efficiency because its saying market return does not adjust immediate;
asset characteristic line
instead of risk vs return line, its market excess return vs asset excess return line (Rm-Rf vs Ri-Rf);
slope is beta=asset return over market return;
slope of any regression line=covariance of two variables divide by variance of individual term but because this is relationship over market return and covariance is unaffected by the rf asset so denominator is market square (correlation=Cov (i,m) / σ(m)^2)=relative systematic risk (covariance is stock’s systematic risk / market systematic risk which is itself);
!!!! βi=Cov (i,m) / σ(m)^2);
portfolio β=weighted averaged of individual beta;
plug in correlation:
!!!! βi=cor(i,m) * σ(i) / σ(m);
βm=cor(m,m) * σ(m) / σ(m)=1 cuz beta here is relative systematic risk;
M^2
M^2=CAL portfolio return if the risk is scaled up to same as market line CML;
M^2=Rf+(σm/σp)*(Rp-Rf);
if M^2 return=CML return, portfolio is fairly priced, if >, superior portfolio;
M^2 alpha=M^2-Rm=excess return for a leveraged portfolio with same risk as market portfolio; measures total risk-adjusted performance (RAP)
treynor measure P
treynor measure=(Rp-Rf)/βp;
excess return between “CAL” and SML line risk adjusted by per unit of x axis systematic risk of port βp;
similar to sharpe but change total risk to systematic risk with beta market=1;
higher good;
jensen’s alpha=pf-return of CAPM=excess return above equilibrium return for a portfolio with beta
Treynor measure is excess return (return in excess of the risk-free rate) per unit of systematic risk (beta) [CAPM].
The Sharpe ratio is excess return per unit of total risk (portfolio standard deviation) [CAL, CML].
M^2 is alternative to sharpe in % use total risk [CAL, CML].
Jensen’s alpha is the difference between a portfolio’s ACTUAL rate of return and the equilibrium rate of return for a portfolio with the same level of beta (systematic) risk
treynor=(Rp-Rf)/βp;
sharpe=(Rp-Rf)/σp;
M^2=Rf+(σm/σp)(Rp-Rf)=sharpeσm+Rf; if positive, manager added value;
M^2 alpha=M^2-Rm;
jensen’s alpha=Rp-return of CAPM; if positive, manager added value;
CAL expected return=Rf + [(E(Rm) – Rf)/σm] × σp
Total risk equals systematic plus unsystematic risk. Unique risk is diversifiable and is unsystematic. Market (systematic) risk is nondiversifiable risk.
When you increase the number of stocks in a portfolio, unsystematic risk will decrease at a decreasing rate. However, the portfolio’s systematic risk can be increased by adding higher-beta stocks or decreased by adding lower-beta stocks
portfolio with a rf asset variance and sd
Var(Rp)=σA^2WA^2
σ(Rp)=σAWA
CML, the market portfolio includes:
contains all risky assets in existence. It does not contain any risk-free assets.
When the market is in equilibrium, expected returns equal required returns. Since this means that all assets are correctly priced, all assets plot on the SML.
By definition, all stocks and portfolios other than the market portfolio fall below the CML. (Only the market portfolio is efficient.)
