Estimating Portfolio Return using History

In this post we want to assess the returns of the portfolio using the History.

Portfolio

           Portfolio is a  group of financial assets such as stocks, bonds and cash equivalents held by investors. Portfolio can be explained as below:-
                  1) Portfolio members      =  I  = {I1  ,I2  ,...In}.
                  2) Portfolio distribution  = D  = {D1,D2,..Dn},(D1+D2+..+Dn=1)

                  3) Portfolio value            = V  = I*D  

Portfolio Returns Assessment

         
        Guidance :-
           large changes tend to be followed by large changes-of either sign-and small changes                          by small changes(Mandelbrot 1963).

        Solution:-
            Obtain how current portfolio would have performed in given period by doing the following:-
             

						PortfolioHistory							PortInformation
							I1	I2	I3					I1	I2	I3	PortfolioVALUE	PortfolioRETURN
						01/01/01	2.3	2	3				01/01/01	1.15	0.6	0.6	2.35	.
						01/02/01	4	5	6				01/02/01	2	1.5	1.2	4.7	1
	CurrPortfolioDistribution					01/03/01	2	3	4				01/03/01	1	0.9	0.8	2.7	-0.425532
	0.5	0.3	0.2		*	01/04/01	2.1	2	3		=		01/04/01	1.05	0.6	0.6	2.25	-0.166667
						01/05/01	2.5	2.2	3.1				01/05/01	1.25	0.66	0.62	2.53	0.1244444
						01/06/01	1.95	2.22	3.11				01/06/01	0.975	0.666	0.622	2.263	-0.105534
						01/02/15	1.95	2.22	3.11				01/02/15	0.975	0.666	0.622	2.263	-0.105534

          Statistics:-

Moments
N	5000000	Sum Weights	5
Mean	0.08534245	Sum Observations	0.42671227
Std Deviation	0.54750851	Variance	0.29976557
Skewness	1.54600233	Kurtosis	2.75815662
Uncorrected SS	1.23547895	Corrected SS	1.19906228
Coeff Variation	641.542972	Std Error Mean	0.24485325

Quantiles (Definition 5)
Level	Quantile
100% Max	1.000000
99%	1.000000
95%	1.000000
90%	1.000000
75% Q3	0.124444
50% Median	-0.105534
25% Q1	-0.166667
10%	-0.425532
5%	-0.425532
1%	-0.425532
0% Min	-0.425532

           Var with 95% is 0.425532 so the loss will not be more than 42.5532% (Too high right, blame it on our sample data).

Variations:-

    1) Windowed approach (i.e consider only past K returns for VaR calculation with smaller window           size quickly adapt to volatility & larger window of high precision).

As per the study of historical data with parametric assumption we found the VaR. There are better 

alternatives, we can see about the same in future posts.

Better Alernatives:-

   

   1) RiskMetrics Exponential Smoothing [Places exponentially declining weights on historical data,            starting with an initial weight, and then declining to zero as we go further into the past].

   2) Adaptive Volatility Estimation.

   3) GARCH

   4) Multivariate Density Estimate(which i like the most!!).

Thanks 

Srinivas