SOCR EduMaterials Activities ApplicationsActivities Portfolio

From Socr

Revision as of 06:02, 3 August 2008 by Nchristo (Talk | contribs)
Jump to: navigation, search

Portfolio Theory

An investor has a certain amount of dollars to invest into two stocks (IBM and TEXACO). A portion of the available funds will be invested into IBM (denote this portion of the funds with xA) and the remaining funds into TEXACO (denote it with xB) - so xA + xB = 1. The resulting portfolio will be xARA + xBRB, where RA is the monthly return of IBM and RB is the monthly return of TEXACO. The goal here is to find the most efficient portfolios given a certain amount of risk. Using market data from January 1980 until February 2001 we compute that E(RA) = 0.010, E(RB) = 0.013, Var(RA) = 0.0061, Var(RB) = 0.0046, and Cov(RA,RB) = 0.00062. We first want to minimize the variance of the portfolio. This will be: 
\mbox{Minimize} \ \ \mbox{Var}(x_A R_A+x_BR_B) 
\mbox{subject to} \ \ x_A+x_B=1
Or 
\mbox{Minimize} \ \ x_A^2 Var(R_A)+x_B^2 Var(R_B) + 2x_Ax_BCov(R_A,R_B) 
\mbox{subject to} \ \ x_A+x_B=1
Therefore our goal is to find xA and xB, the percentage of the available funds that will be invested in each stock. Substituting xB = 1 − xA into the equation of the variance we get 
x_A^2 Var(R_A)+(1-x_A)^2 Var(R_B) + 2x_A(1-x_A)Cov(R_A,R_B).
To minimize the above exression we take the derivative with respect to xA, set it equal to zero and solve for xA. The result is: 
x_A=\frac{Var(R_B) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
and therefore 
x_B=\frac{Var(R_A) - Cov(R_A,R_B)}{Var(R_A)+Var(R_B)-2Cov(R_A,R_B)}
The values of xa and xB are: 
x_a=\frac{0.0046-0.0062}{0.0061+0.0046-2(0.00062)} \Rightarrow x_A=0.42.
and x_B=1-x_A=1-0.42 \Rightarrow x_B=0.58. Therefore if the investor invests 42 \% of the available funds into IBM and the remaining 58 \% into TEXACO the variance of the portfolio will be minimum and equal to: Var(0.42RA + 0.58RB) = 0.422(0.0061) + 0.582(0.0046) + 2(0.42)(0.58)(0.00062) = 0.002926. The corresponding expected return of this porfolio will be: E(0.42RA + 0.58RB) = 0.42(0.010) + 0.58(0.013) = 0.01174. We can try many other combinations of xA and xB (but always xA + xB = 1) and compute the risk and return for each resulting portfolio. This is shown in the table and the graph below.

xA xB Risk (σ2) Return Risk (σ)
1.00 0.00 0.006100 0.01000 0.078102
0.95 0.05 0.005576 0.01015 0.074670
0.90 0.10 0.005099 0.01030 0.071404
0.85 0.15 0.004669 0.01045 0.068329
0.80 0.20 0.004286 0.01060 0.065471
0.75 0.25 0.003951 0.01075 0.062859
0.70 0.30 0.003663 0.01090 0.060526
0.65 0.35 0.003423 0.01105 0.058505
0.60 0.40 0.003230 0.01120 0.056830
0.55 0.45 0.003084 0.01135 0.055531
0.50 0.50 0.002985 0.01150 0.054635
0.42 0.58 0.002926 0.01174 0.054088
0.40 0.60 0.002930 0.01180 0.054126
0.35 0.65 0.002973 0.01195 0.054524
0.30 0.70 0.003063 0.01210 0.055348
0.25 0.75 0.003201 0.01225 0.056580
0.20 0.80 0.003386 0.01240 0.058193
0.15 0.85 0.003619 0.01255 0.060157
0.10 0.90 0.003899 0.01270 0.062439
0.05 0.95 0.004226 0.01285 0.065005
0.00 1.00 0.004600 0.01300 0.067823

For the above calculations short selling was not allowed (0 \le x_A \le 1 and 0 \le x_B \le 1, in addition to xA + xB = 1). We note here that the efficient portfolios are located on the top part of the graph between the minimum risk portfolio point and the maximum return portfolio point, which is called the efficient frontier (the blue portion of the graph). Efficient portfolios should provide higher expected return for the same level of risk or lower risk for the same level of expected return.

If short sales are allowed, which means that the investor can sell a stock that he or she does not own the graph has the same shape but now with more possibilities. The investor can have very large expected return but this will be associated with very large risk. The constraint here is only Failed to parse (lexing error): x_A+x_B=1<math>, since either <math>x_A<math> or <math>x_B<math> can be negative. The snapshot below from the SOCR applet shows the ``short sales scenario" for the IBM and TEXACO stocks. The blue portion of the portfolio possibilities curve occurs when short sales are allowed, while the red portion corresponds to the case when short sales are not allowed. <br> When the investor faces the efficient frontier when short sales are allowed and he or she can lend or borrow at the risk-free interest rate the efficient frontier will change in the following way: Let <math>x

be the portion of the investor's wealth invested in portfolio A that lies on the efficient frontier, and 1 − x the the portion invested in a risk-free asset.  This combination is a new portfolio and has


\bar R_p=x\bar R_A + (1-x)R_f
where Rf is the return of the risk-free asset. The variance of this combination is simply 
\sigma_p^2=x^2 \sigma_A^2 \Rightarrow x=\frac{\sigma_p}{\sigma_A}
From the last two equations we get 
\bar R_p = R_f + \left(\frac{\bar R_A-R_f}{\sigma_A}\right)\sigma_p


This is an equation of a straight line. On this line we find all the possible combinations of portfolio A and the risk-free rate. Another investor can choose to combine the risk-free rate with portfolio B or portfolio C. Clearly, for the same level risk the combinations that lie on the RfB line have higher expected return than those on the line RfA < math > (seefigurebelow).And < math > RfC will produce combinations that have higher return than those on RfB for the same level of risk, etc.

Personal tools