A. Caggia – M. Armanini Financial Investment & Pricing 2016-2017 Ref: Yale University Publications Investment and Portfolio Management, Bodie Kane and Marcus Robert Shiller, Financial Markets Web sources Portfolio Management A. Caggia – M. Armanini Financial Investment & Pricing 2016-2017

Definitions Portfolio = is a set of financial instruments Asset Allocation = is the decision by the investor of the types and amounts of assets to assign to a portfolio Portfolio Management = is the activity of constructing and following the evolution of a portfolio during its lifetime, by balancing risk and reward

Portfolio Composition Equities Bonds Currency Commodities Index

Returns Single Asset Return (R) = is the percentage increase (or decrease) in the value of the asset during a period of time (e.g. one week)

First a quick refresh X,Y = two random variable expected value, variance, covariance and correlation In our case the random variables are the returns

Expected Value, Mean, Average

Returns Portfolio Return (Rp) = is the percentage increase (or decrease) in the value of the portfolio during a period of time (e.g. one week) Two assets: N assets:

Variance Var(x + y) = var(x) + var(y) + 2 cov(x,y) Covariance

Correlation Coefficient A scaled measure of how much two variables move toghether -1 ≤ r ≤ 1 rx,y = cov(x,y) / (sxsy)

Portfolio Selection (1952) Harry Markowitz (1927) in 1952 (age 25) published an article that changed the way we look at risk. Nobel price 1990 …

A Portfolio of a Risky and a Riskless Asset Put x euro in risky asset 1, (1-x) euro in the riskless asset earning a sure return rf Portfolio expected value r = xr1 + (1 – x)rf Portfolio variance = x2 var(r1)

A Portfolio of 2 Risky Assets Put x1 euro in risky asset 1 and (1-x1) euro in risky asset 2 Portfolio expected value r = x1r1+(1-x1)r2 Portfolio variance = X21var(return1 )+ (1− x1 )2 var(return2 )+ 2x1 (1−x1 )cov(return1,return2 )

Efficient Portfolio Frontier Stocks and Bonds

Portfolio expected returns as a function of weights If wd=1.5 and we = -0.5 (short equity long fund bonds) Expected return = 5.5%

Portfolio Std Dev as a function of weights With R=.30, std dev first falls for diversification benefit, then it grows because equity weights more and diversification benefits are decreasing

Portfolio expected returns as a function of std dev R=1 perfectly correlated, NO diversification benefits R=-1 perfectly negatively correlated MAX diversification benefit could reach 9,875% with zero std dev. Weights wd=0,625, we=0,275 R=0 no correlation min portfolio std dev 10,29% expected return 9,32% R=0,30 exp return 8,9% std dev 11,44

Important to notice There are portfolios where the std dev is lower than either of the two components (e.g. min variance portfolio) Potential benefits from diversification are greater if correlation is less then perfectly positive Decisions on where to position on the efficient frontier is risk aversion linked

Efficient Portfolio with and without Oil

Sharpe Ratio for a Portfolio The Sharpe Ratio is constant along the tangency line A portfolio manager is outperforming only if his portfolio has a greater Sharpe ratio