Measures of dispersion of actual return from expected return:
variance (defined as average square ‘uncertainties’)(σ2) or standard deviation (square root of variance)(σ)
Measure the total risk:
Total Risk = Market (or systematic) + Company Specific (or unsystematic)
Standard Deviation
1 σ
2 σ
3 σ
1 σ = 68.5%
2 σ = 95.4%
3 σ = 99.7%
Skewness
Distribution can be positive or negative
Normal distribution skewness is 0
In a large enough portfolio with normal distribution, the skewness will cancel each other out.
There are also mathematical transformations that can be applied, by adding 1 to Ln to ‘normalise’ it.
Kurtosis
Kurtosis measures the degree of ‘fat-tails’ in a distribution.
• Risk averse investor prefers low kurtosis.
• Normal distribution kurtosis is 3.
‘Fat-tailed’ distribution: tendency of asset’s prices and return distributions to have more observations in the tails and to be thinner in the midrange than a normal (bell shaped curve) distribution
Leptokurtic
A distribution with positive excess kurtosis is called leptokurtic
In terms of shape, a leptokurtic distribution has a more acute peak around the mean and fatter tails.
Platykurtic
A distribution with negative excess kurtosis is called platykurtic
In terms of shape, a platykurtic distribution has a lower, wider peak around the mean and thinner tails.
Semi Deviation (σD) and Semi Variation(σD2)
Used by investors who are interested in downside risk only.
Semi-deviation is a square root of semi-variance
Semi-variance is an average of the squared deviations of values that are less than the mean return or target value.
Converting prices —> Returns
Geometric Rate of Return (continuously compounded return)
Rt = ln [Pt+Dt-1]/Pt-1
Variance & Covariance
Covariance shows the relationship between two factors, and this can be positive or negative.
V(Z) = V(aX ) + V(bY) +_ 2Cov(aX,bY) = a2V(X ) + b2V(Y) +_ 2abCov(X,Y)
Where the term Cov(X ,Y ) is called the covariance between random variables X and
Y. The formula for the covariance is:
Cab = ∑n hi [Rai - E(Ra)][Rbi - E(Rb)]
i=1
hi = probabilities.
Formula for Variance
The covariance of the asset with itself is the variance of the asset.
Va = ∑hi(Ra-E(Ra))2
Sum of probabilities of Return of Asset A - Expected Return of Asset A squared.
Two Asset Portfolio: Expected Return and Risk
Consider an investor that has two assets, A and B that are components part of his/her portfolio. If proportion of total investment in asset A is 1 w and in asset B is 1- 1 w , then
the return on a two-asset portfolio is given by:
E(rp) = w1E(r1)+w2E(r2) since w1+w2 = 1 w2 = 1-w1 then E(rp) = w1E(r1)+(1-w1)E(r2)
Variance on two asset portfolio V(P) = σ2 p V(A) = σ2 1 V(B) = σ2 2 Cov(A,B) = σ1,2
σ2 = w2σ2+(1-w1)2σ2+2w1(1-w1)σ1,2
p 1 1 2