Understanding Portfolio Theory: Overview & Formulas
Portfolio theory as developed by Harry Markowitz is based on four asumptions:
- Investors have a single investment horizon
- Each security is represented by a probability distribution of expected rate of return
- A portfolio of individual securities is completely described by the expected portfolio return and variance of the portfolio return.
- Investors are single period expected utility maximizers. For any given level of risk, investors prefer more return to less and for any given return investors prefer less risk to more.
There are no transaction costs and the model measures correlation for every pair of shares.
There are two phases in formulating Markowitz model: Security Analysis & Portfolio Analysis
1. Security Analysis
The individual securities are analyzed by estimating:
- A single period return on a security
- The expected return and variance of the rate of return
- The covariance and correlation coefficient between rates of return
Single period return on a security:
Rt= Pt- Pt-1 + Dt
Pt-1 Pt-1
Since the expected price and dividends per share are uncertain, the rate of return is also uncertain. Thus, the sensitivity analysis of the rate of return can be carried out by simulating different expected price and dividend values.
n
Expected return= E(Ri)= ∑ PtRit
t=1
n
Variance of a security’s return(σ2) = ∑ Pt( Rit – E(R)) 2
t=1
The tighter the distribution of returns, lower the standard deviation, the lower will be the risk of an investment.
Coefficient of variation= σj
μj
The coefficient of variation shows the risk per unit of return and it provides a more meaningful basis for comparison when the expected returns on two alternatives are not the same.
The covariance of rates of return shows only the direction of the association between the two variables.It does not provide any information about the power of the relationship between the two variables. The correlation coefficient is a better indicator than covariance of the relationship between returns of securities.
n
Cov (Ri,Rj) = ∑ Pk (Rik -E(Ri) ( Rji- E(Rj)
k=1
rij = Cov (Rj,Ri)
2. Portfolio Analysis
The estimations shown above considers the risk of the assets held in isolation.In the portfolio analysis, the risk and return of an individual security is analyzed in terms of how that security affects the risk and return of the portfolio in which it is held.
Portfolio Returns:
The expected return on a portfolio is the weighted average of the expected returns on the individual assets in the portfolio, with the weights being the fraction of the total portfolio invested in each asset:
Rp = ∑ wiĵ ri
Portfolio Risk:
The risk of a portfolio is not a weighted average of the standard deviation of individual securities in the portfolio—the portfolio risk will be smaller because variations in stocks’ returns will be offsetting to some degree.
However, diversification will be of no use in reducing risk, if the portfolio consists of perfectly positively correlated stocks (ρ =+1)
Theoretically, it is possible to reduce portfolio risk to zero by forming a portfolio with perfectly negatively correlated stocks.( ρ= -1).In this case, the risk is totally diversified away.
When correlation coefficient for returns of two stocks is above zero but less than +1 , combining such stocks into portfolios reduces risk but does no eliminate it completely.
n n n
σ2 (Rp) =∑ Xi2 σi2 + ∑ ∑ XiXj Cov (Ri, Rj)
i=1 i=1 j=1 .
i≠j
The N security portfolio requires ( N2-N )/ 2 covariance values.For example, if n=30, σp requires 435 covariance values.
Two asset case:
m m
σp = ( ∑ ∑ Xi Xj rjk σj σk )1/2
j=1 k=1
j ≠ k
σ2 = Xi2σi2 + 2 XiXj rjk σj σk + Xj σj2
Some value of X will minimize the portfolio standard deviation.To obtain this value we differentiate the equation with respect to X1 and and set thederivative equal to zero.The optimal weight for security 1 is:
X1= σ 2(σ2 –r12σ1)
σ12+ σ22 -2r12 σ1σ2
= σ22 –Cov 12
σ12+ σ22-2Cov 12
The optimal weight for security 2 can be estimated from 1- X1
The Bottom Line
Diversification across industries deals with investing in a number of randomly selected securities from different industries. Evidence indicates that diversification across industries is not much better than naive diversification because rates of return on securities from different industries are often highly correlated.
Naive diversification and Markowitz diversification are two different methods for reducing portfolio risk.The former does not guarantee a minimum level of risk at a specific level of return.
In diversification across industries, the unsystematic risk decreases fast as more securities are added to the portfolio but at a decreasing rate.Total risk declines as the number of securities held increases. The systematic risk is not reduced , so total portfolio risk is reduced but not eliminated.
Empirical studies indicate that risk is substantially reduced in portfolios containing 10-15 shares as compared with the risk of holding one share. When market value weights are used, a greater number of securities is required to reduce the risk by a comparable amount.
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