Portfolio theory
Portfolio theory
DESCRIPTIVE STATISTICS
•
From prices to returns
•
Mean => =AVERAGE(X)
•
Variance => =VARP(X)
•
Standard deviation => =STDEVP(X)
Covariance and correlation
•
Degree to which the returns on the two assets
move together:
•
Covariance => =COVAR(X;Y)
•
Degree of linear relation between reurns:
•
Correlation => =CORREL(X;Y)
(unit-free)
Add Trendline
Portfolios: 2 assets
•
Mean:
•
Variance:
Portfolios: n assets
•
Mean:
•
Variance:
•
Covariance:
Portfolios of 4 assets
Variance-Covariance matrix
•
Additional returns Matrix:
•
n=number of assets
•
m=number observations
•
Variance covariance matrix:
Minimun variance portfolios
•
Every portfolio on the minimum variance frontier:
•
R – c = S * z
•
R = vector of E(R
i
)
•
c = constant
•
S = variance-covariance matrix
•
z = portfolio components
•
x = normalized portfolio components:
•
Select two values for c (correspond to two
portfolios):
•
z = S
-1
* ( R – c )
•
compute x
i
Minimum variance portfolios
WHEN c = rf, THE ENVELOPE PORTFOLIO
IS THE MARKET PORTFOLIO M
In the realm of finance, portfolio theory plays a crucial role in investment decisions by analyzing the relationship between assets and returns. Descriptive statistics, on the other hand, provide valuable insights from data regarding means, variances, and standard deviations. Covariance and correlation aid in understanding how assets move together and the linear relation between returns. Dive into the world of portfolios with 2 assets or n assets, analyzing mean returns and variances to optimize investment strategies. Explore the concept of minimum variance portfolios and the variance-covariance matrix to construct efficient portfolios. Unveil the essence of the minimum variance frontier, normalized portfolio components, and optimal values for portfolio construction.
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Presentation Transcript
DESCRIPTIVE STATISTICS From prices to returns P = ln t r t P 1 t Mean => =AVERAGE(X) Variance => =VARP(X) Standard deviation => =STDEVP(X)
Covariance and correlation Degree to which the returns on the two assets move together: Covariance => =COVAR(X;Y) Degree of linear relation between reurns: Correlation => =CORREL(X;Y) (unit-free)
Add Trendline 25.00% y = 0.4091x - 0.0019 R = 0.3008 20.00% 15.00% 10.00% 5.00% Series1 Linear (Series1) 0.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% -5.00% -10.00% -15.00% -20.00%
Portfolios: 2 assets Mean: = + ( ) ( ) 1 ( ) ( ) E r E r E r P a b Variance: = + + 2 ) 2 2 2 2 2 1 ( ) 1 ( , p a b a b a b
Portfolios: n assets Mean: n = = T ( ) ( ) ( ) E r E r E r P i i = 1 i = i 2 = T Variance: S p i j ij j T = Covariance: ) 2 , 1 ( Cov S 1 2
Portfolios of 4 assets Portfolios x, y and the Four Stocks 15% 13% Mean return 11% 9% 7% Stock A 5% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% Standard deviation
Variance-Covariance matrix Additional returns Matrix: n=number of assets m=number observations . . . . r r r r 1 n 11 1 n . = A . . . . . . . . . r r mn r r 1 n 1 m Variance covariance matrix: T A A = = [ ] S , i j M
Minimun variance portfolios Every portfolio on the minimum variance frontier: R c = S * z R = vector of E(Ri) c = constant S = variance-covariance matrix z = portfolio components
Minimum variance portfolios x = normalized portfolio components: z x = i i i z i Select two values for c (correspond to two portfolios): z = S-1 * ( R c ) compute xi
WHEN c = rf, THE ENVELOPE PORTFOLIO IS THE MARKET PORTFOLIO M Efficient Frontier with CML Capital market line, CML Portfolio mean return Market Risk-free rate, rf Portfolio standard deviation
