Portfolio Management
The Investment Decision
 Top-down process with 3 steps:
1. Capital allocation: risky portfolio and risk-free asset
2. Asset allocation: across broad asset classes
3. Security selection: individual assets within asset class
Diversification and Portfolio Risk
 Market risk
 Marketwide risk sources
 Remains even after diversification
 Also called: Systematic or Nondiversifiable
 Firm-specific risk
 Risk that can be eliminated by diversification
 Also Called: Diversifiable or Nonsystematic
Portfolio Risk and the Number of Stocks in the
Portfolio
Portfolios of Two Risky Assets
 Expected Return:
 Portfolio risk:
 Covariance:
Portfolios of Two Risky Assets: Expected Return
E (rp )  w D E (rD )  wE E (rE )
Portfolios of Two Risky Assets: Risk
Portfolio variance = w12σ12 + w22σ22 + 2w1w2Cov1,2
Where:
 w1 = the portfolio weight of the first asset
 w2 = the portfolio weight of the second asset
 σ1= the standard deviation of the first asset
 σ2 = the standard deviation of the second asset
 Cov1,2 = the covariance of the two assets, which can thus be
expressed as p(1,2)σ1σ2, where p(1,2) is the correlation coefficient
between the two assets
Portfolios of Two Risky Assets: Covariance
Cov(rD , rE )  rDE D E
Where
 r(D,E) = is the correlation coefficient between the two assets
 σD = Standard Deviation of D security
 σE = Standard Deviation of E security
Correlation: Why Diversification Works!
 Assets that are less than perfectly positively correlated tend to
offset each others movements, thus reducing the overall risk in
a portfolio
 The lower the correlation the more the overall risk in a portfolio
is reduced
 Assets with +1 correlation eliminate no risk
 Assets with less than +1 correlation eliminate some risk
 Assets with less than 0 correlation eliminate more risk
 Assets with -1 correlation eliminate all risk
Combining Negatively Correlated Assets to Diversify
Risk
Risk return in three securities case
Modern Portfolio Theory (MPT)
 Emphasizes statistical measures to develop a portfolio plan
 Focus is on:
 Expected returns
 Standard deviation of returns
 Correlation between returns
 Combines securities that have negative (or low-positive)
correlations between each other’s rates of return
Key Aspects of MPT: Efficient Frontier
 Efficient Frontier
 The leftmost boundary of the feasible set of portfolios that include all
efficient portfolios: those providing the best attainable tradeoff
between risk and return
 Portfolios that fall to the right of the efficient frontier are not
desirable because their risk return tradeoffs are inferior
 Portfolios that fall to the left of the efficient frontier are not available
for investments
The Feasible or Attainable Set and the Efficient
Frontier
Exampe -Portfolios of Two Risky Assets: Example
— 50%/50% Split
Expected Return
E (rp )  w D E (rD )  wE E (rE )
 .50  8%  .50 13%  10.5%
Variance:
 p2  wD2  D2  wE2 E2  2wD wE Cov  rD , rE 
 .502 122  .502  202  2  .5  .5  72  172
 P  172  13.23%
Computation of Portfolio Variance from the
Covariance Matrix
Thus, to know the standard deviation of Portfolio we have to
know:
1. Proportion of funds devoted to each stock
2. Standard Deviation of each stock
3. Covariance between the two stocks
• If stocks are independent of each other:
rxy = 0, the last term would be 0
• If rxy > 0, S.D would be greater than when rxy = 0
• If rxy <0, S.D would be less than when rxy >= 0
The Sharpe Ratio
 Maximize the slope of the CAL for any possible portfolio, P
 The objective function is the slope:
Sp 
E rp   rf
p