Risk and return for a portfolio of 2 risky assets

Expected return of a two asset portfolio

E(R)= w1 x E(R1) + w2 x E(R2).

Variance of a two-asset portfolio
= w12 x s12 + w22 x s22 + 2 x w1 x w2 x s1 x s2 x Cor(1,2)

The expected return of a portfolio is simply sum of weighted returns of each assets in the portfolio. However, the standard deviation of the portfolio should take the correlation of the assets into account.

Covariance and Correlation

Covariance (Cov)
Cov(1,2) = Σ {[Ri,1 – mean(R1)][ Ri,2 – mean(R2)]}/(n-1)

Cov(1,2) = Σ Pi[Ri,1 – mean(R1)][ Ri,2 – mean(R2)]


Correlation (Cor)
To standardize covaraince, use correlation:

Cor (1,2) = Cov(1,2) / (σ 1 x σ 2)

The correlation coefficient is the relative measure of the relationship between two assets. It is between +1 and -1, with a +1 indicating that the two assets move completely together and a -1 indicating that the two assets move in opposite directions from each other.

Notes:
The correlation between investment grade indexes is greater than the correlation between high yield indexes ; Low correlation between monthly country equity index result in gain from global diversification.

Variance and Standard Deviation

Variance (σ ²) = Σ[Ri-E(R)]²xPi

Standard Deviation (σ )=(σ ²)^1/2

Expected Return E(R)

The expected return for an individual investment:

E(R) = p1R1 + p2R2 + …..+ pnR
Where:
pn = the probability the return actually will occur in state n
Rn = the expected return for state n

The expected return on a portfolio:
E(R) of a portfolio = w1R1 + w2R2 + …+ wnRn

Where:
Wi = weight of asset i
Ri = the expected return of asset i

Indifference curve

Indifference curve (expected return vs Expected risk):
For investors, return and risk were the key objectives. An investor's risk profile is illustrated with indifference curves. The optimal portfolio, then, is the point on the efficient frontier that is tangential to the investor's highest indifference curve.

Steep indifference curve=> convervative investor;
Flatter indifference curve=>less risk averse investor
Higher curve =>greater utility; more utility is preferable

Markowitz portfolio theory

Investors base investment decisions on expected risk and return, and prefer higher returns to lower returns and lower risk to higher risk.

Assumptions:
· Prefer lower risk for the same level of expected return
· Risk in terms of an investment's variance or standard deviation.
· Investment expected return and probability of the returns over a period is quantifiable
· Investors make decision based on an investment's risk and return, therefore, an investor's utility curve is based on risk and return.

Markowitz’s efficient frontier
The curve represents the set of portfolios that have the highest expected return for a given level of risk and the least risk for a given level of expected return (in terms of standard deviation)

While an efficient frontier illustrates each of the efficient portfolios relative to risk and return levels, each of the efficient portfolios may not be appropriate for every investor.

Risk Aversion

Investor prefers:
· higher return for same risk
· lower risk for same return