1. 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

2. 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

3. Portfolio Composition
Equities
Bonds
Currency
Commodities
Index

4. 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)

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

6. Expected Value, Mean, Average

7. 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:

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

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

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

11. 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)

15. 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 )

16. Efficient Portfolio Frontier Stocks and Bonds

18. 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%

19. 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

20. 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

21. 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

22. Efficient Portfolio with and without Oil

23. 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