Portfolio analysis

Example. Historical Annual Volatilities and Correlations for Selected Stocks
• Stock returns tend to move together if they are affected similarly by economic events (same industry vs. different industry)
Finance course – Prof. Samuele Murtinu
Covariance and Correlation
• Covariance " The expected product of the deviations of two returns from their means " Covariance between the returns Ri and Rj
Value of investment i xi = Total value of portfolio
• Return on the portfolio (Rp): weighted average of the returns on the investments in the portfolio [weights correspond to portfolio weights]
Finance course – Prof. Samuele Murtinu
The Expected Return of a Portfolio
MARKOWITZ’s MODEL (1952) • Optimal portfolio ! method to define a portfolio and its return • Portfolio Weights: fraction of the total investment in the portfolio held in each individual investment in the portfolio • Portfolio weights must add up to 1 (or 100%)
Finance course – Prof. Samuele Murtinu
Textbook Example 11.1
Finance course – Prof. Samuele Murtinu
The Expected Return of a Portfolio
• The expected return of a portfolio is the weighted average of the expected returns of the investments within it
• If the covariance is positive, the two returns tend to move together (above or below average at the same time) • If the covariance is negative, the two returns tend to move in opposite directions • Sign is easy to interpret BUT the magnitude is not!
Textbook Example 11.6
• The variance of the portfolio depends not only on the variance of the individual stocks, but also on the covariance between them • The more the stocks move together, the higher their covariance or correlation, the more variable the portfolio will be
Corr (Ri ,R j ) =
Cov(Ri ,R j ) SD (Ri ) SD(R j )
" The correlation between two stocks will always be between –1 and +1 " Same sign of covariance
Finance course – Prof. Samuele Murtinu
2 Var (RP ) = x12Var (R1 ) + x2 Var (R2 ) + 2 x1 x2Cov(R1 ,R2 )
• Volatility: square root of the variance ! SD(RP)=√Var(RP)
Finance course – Prof. Samuele Murtinu
RP = x1 R1 + x2 R2 + L + xn Rn =
Finance course – Prof. Samuele Murtinu
∑ xR
i i
i
Textbook Example 11.1
• Initial weights: • xW=200*30/10000=6000/10000=60% • xC=100*40/10000=4000/10000=40%
E [ RP ] = E ∑ i xi Ri =
∑ E[x R ]
i i i
=
∑ x E [R ]
i i i
Finance course – Prof. Samuele Murtinu
The Volatility of a Two Two-Stock Portfolio
Combining stocks ! part of the risk is eliminated through diversification
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Portfolio theory
Prof. Samuele Murtinu
Finance course – Prof. Samuele Murtinu
Agenda
• Markovitz’s model Expected Return of a Portfolio Volatility of a Two-Stock Portfolio Volatility of a Large Portfolio Risk Versus Return: Choosing an Efficient Portfolio
Finance course – Prof. Samuele Murtinu
Covariance and Correlation
• Correlation " A measure of the common risk shared by stocks that does not depend on their volatility
Finance course – Prof. Samuele Murtinu
The Volatility of a Two Two-Stock Portfolio
• First portfolio: equal investments in the two airlines • Second portfolio: equal investments in West Air and Tex Oil ! Average return of the portfolio is equal to the average return of the two stocks ! Volatility of 5.1% is much less than the volatility of the two individual stocks • Both portfolios have lower risk than the individual stocks
Cov(Ri ,R j ) = E[(Ri − E[ Ri ]) (R j − E[ R j ])]
" Estimate of the Covariance from Historical Data
Cov(Ri ,R j ) =
1 (Ri ,t − Ri ) (R j ,t − R j ) ∑ t T − 1
• To find the risk of a portfolio, we must know the degree to which the stocks’ returns move together: " Covariance " Correlation Allow us to measure the co-movement of returns
The 3 stocks have the same volatility and average return, BUT the pattern of their returns differs ! when the airline stocks performed well, the oil stock tended to do poorly, and viceversa
SD(RP ) =
2 x12Var (R1 ) + x2 Var (R2 ) + 2 x1 x2Cov(R1 ,R2 )
Finance course – Prof. Samuele Murtinu
Finance course – Prof. Samuele Murtinu
The Volatility of a Two Two-Stock Portfolio
• The amount of risk that is eliminated in a portfolio depends on the degree to which the stocks face common risks and their prices move together " The two airlines tend to perform well or poorly at the same time (first portfolio has a volatility that is only slightly lower than that of the individual stocks) " Airline and oil stocks do not move together BUT they move in opposite directions (second portfolio is MUCH LESS risky)
• Tobin’s model - Risk-Free Saving and Borrowing - The Efficient MARKET Portfolio and Required Returns • Capital Asset Pricing Model (CAPM) - Determining the Risk Premium
Var (RP ) = Cov(RP ,RP ) = Cov(x1 R1 + x2 R2 ,x1 R1 + x2 R2 ) = x1 x1Cov(R1 ,R1 ) + x1 x2Cov(R1 ,R2 ) + x2 x1Cov(R2 ,R1 ) + x2 x2Cov(R2 ,R2 )
• The Variance of a Two-Stock Portfolio
Finance course – Prof. Samuele Murtinu
Textbook Example 11.5
Finance course – Prof. Samuele Muance and Volatility
• The variance of a return is equal to the covariance of a return with itself • For a two-stock portfolio with RP=x1*R1+x2*R2: