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

VaR BackTesting

VaR Analysis:-

One of widely used risk metric in financial industry is VaR(Value At Risk) .VaR is a risk measure which will give a magical number about loss (for a given period D with probability p% loss will not exceed x%).

For effective backtesting, VaR should exhibit following desirable characters:-
                 1)  Unbiasedness   [Average of Indicator Variable should be x%].
                  2) Don'tBunchUp [VaR estimates do not bunch up over a particular state of economy].

Verification:-
Backtesting is used to verify VaR validity[accept or reject], Major concerns here is to balance following types of errors:-

                 1) Type-I Error  [Possiblity that an accurate risk model would be classified as inaccurate].
                 2) Type-II Error [Possiblity that an inaccurate model would not be classified as accurate].

There is no statistical way to handle both Type I & Type II error so Basel came up with zone based approach of handling VaR and deciding Risk Weight.

	Zone	Description	Exceptions Count	Scaling factor Increase	Cumulative Probability
	Green Zone	Test results are consistent with an accurate model, And the possibility of erroneously accepting an Inaccurate model is low.	0	0	8.11%
			1	0	28.58%
			2	0	54.32%
			3	0	75.81%
			4	0	89.22%
	Yellow Zone	Test results are either consistent with either accurate or Inaccurate models, and the supervisor should encourage A bank to present additional information about its model Before taking action.	5	0.4	95.88%
			6	0.5	98.63%
			7	0.65	99.60%
			8	0.75	99.89%
			9	0.85	99.97%
	Red Zone	Test results are extremely unlikely to have resulted From an accurate model, and the probability of Erroneously rejecting an accurate model on this Basis is remote.	10 or more	1	99.99%

SASImplementation

VaRData:-
Data table contains Observations  containing (Date,PredictedVaR and ActualLoss), as below:-

Obs	DATE	PredictedVaR	ActualLoss
1	13894	102232	400000
2	13895	496875	450000
3	13898	406250	393750
4	13899	306250	387500
5	13900	506250	387500
6	13901	506250	475000
7	13902	510938	475000
8	13905	515625	478906

Backtesting Result:-

Using below procedure we get ExceptionCount and Zone in which VaR lies.  

proc IML;

 use Work.VarData;

 read all var {"ActualLoss","PredictedVaR"};

 close Work.VarData;

 indicatorVar             =(ActualLoss>PredictedVar);

 excessLoss               =(ActualLoss-PredictedVar);

 ExceptionCount       = indicatorVar[+,1];

 ExcpetionPercentage=(ExceptionCount/nrow(indicatorVar));

 if ExceptionCount > =10 then do;

   Zone="Red Zone";

 end;

 else do;

    if ExceptionCount <= 4 then do;

      Zone="Green Zone";

    end;

    else do;    

      Zone="Yello Zone";

    end;

 end;

 print Zone ExceptionCount ExcpetionPercentage;

run; 

Zone	ExceptionCount	ExcpetionPercentage
Green Zone	2	0.25

Conclusion:-

After analysis of ActualLoss & PredictedVaR we calculated ExceptionCount and then found the VaR is in Yellow Zone.So not a big problem :). 

Based on 

Thanks & Regards,

Srinivasan

                     

FRTB Partial RiskFactor

Partial RiskFactor:-
       
           In FRTB, partial RiskFactor approach is specified under Revised Standardised Approach as a preferred way of arriving at risk weights. In this approach instruments with similar characters are grouped together in buckets and regulatory prescribed risk weights will be applied to their notional positions.
       
         We have implemented Partial RiskFactor using Clustering procedure of SAS!!! Happy Reading.

Partial RiskFactor Implementation:-
           In SAS CLUSTER can be used, CLUSTER hierarchically clusters the observations in a SAS dataset by using one of 11 methods. PROC CLUSTER computes possibly squared Eucledian distance as part of Clustering.  The methods differ by how distance between two clusters is computed(refer Appendix).

          We have used Wald's method for clustering as it is robust and reliable among the available options.  Please see the portfolio data:-

Input Data:-
 data Portfolio;
      input IReturn IRisk Sector Instrument$20.;
      datalines;
   14.7  5.7  5  Instr1    
   24.7  6.7  4  Instr5
   20.7  5.7  5  Instr9
   24.7  5.7  4  Instr13
   11.5 11.9  3  Instr2
   12.5 12.9  3  Instr10
   :
   ;

Clustering Process:-

To obtain approximate estimate of  pooled within-cluster covariance matrix using aceclus. 
     proc aceclus data=Portfolio out=Ace p=.03  noprint;
         var IReturn IRisk Sector;
     run;

To perform clustering of Instruments use below steps:-
     proc cluster data=Ace method=ward ccc pseudo print=7 outtree=Tree;
        var IReturn IRisk Sector;
         id Instrument;
     run;
     proc tree data=Tree ncl=6 out=clus3;
        copy IReturn IRisk Sector;
     run;

After clustering we can see the hierarchical grouping of instruments and their groups as in below.

Obtaining cluster Membership:-

To obtain cluster & their constituent instruments use the below procedures.

proc sort data=clus3; by cluster;

proc print data=clus3; by cluster;
var IReturn IRisk Sector;
run;

Cluster Analysis of Portofolio                                                                  

CLUSTER=1

Obs	IReturn	IRisk	Sector
1	13.4	11.7	2
2	13.4	11.7	2
3	13.4	11.7	2
4	13.4	11.7	2
5	12.0	12.4	1
6	12.0	12.3	1
7	11.5	11.9	3
8	12.3	11.9	3
9	11.0	12.4	1
10	12.5	12.9	3

CLUSTER=2

Obs	IReturn	IRisk	Sector
11	24.7	6.7	4
12	24.7	5.7	4

CLUSTER=3

Obs	IReturn	IRisk	Sector
13	20.7	5.7	5

CLUSTER=4

Obs	IReturn	IRisk	Sector
14	14.7	5.7	5

CLUSTER=5

Obs	IReturn	IRisk	Sector
15	12.5	1.2	3

CLUSTER=6

Obs	IReturn	IRisk	Sector
16	3	12.4	1

Using the above procedure we can decide which instruments will go into which cluster.

How to assign RiskWeight:-
     Based on Risk&Return attributed of the cluster supervisor will decide that :) .
     Here cluster2,cluster3 will have less riskweights, cluster6 will have more risk weights.

As per your problem choose the required clustering methods( listed in appendix),number of clusters, instrument attributes.

Thanks & Regards
Srinivasan

Appendix
The following are the types of clustering methods:-

     

	Description	Disadvantage
Average Linkage	Distance between two clusters is the average distance between pairs of observations, one in each cluster. Tends to join clusters with small variances.	Slightly biased toward producing clusters with the same variance.
Centroid Method	Distance between two clusters is defined as the (squared) Euclidean distance between their centroids or means. The centroid method is more robust to outliers than most other hierarchical methods.	Might not perform as well as Ward’s method or average linkage.
Complete Linkage	Distance between two clusters is the maximum distance between an observation in one cluster and an observation in the other cluster.	It can be Severely distorted by moderate outliers. Strongly biased toward producing clusters with roughly equal diameters.
Density Linkage	Class of clustering methods that use nonparametric probability density estimates. Density linkage consists of two steps: 1) A new dissimilarity measure, , based on density estimates and adjacencies is computed. 2) A single linkage cluster analysis is performed using dissimilarity analysis is performed. Types are:- kth-Nearest-Neighbor Method Uniform-Kernel Method Wong’s Hybrid Method
Maximum Likelihood	Similar to Ward’s minimum-variance method but removes the bias toward equal-sized clusters.	EML is somewhat biased toward unequal-sized clusters.
Single Linkage	Distance between two clusters is the minimum distance between an observation in one cluster and an observation in the other cluster. Single linkage has many desirable theoretical properties.	Sacrifices performance in the recovery of compact clusters in return for the ability to Detect elongated and irregular clusters. Tends to chop off the tails of distributions before separating the main clusters.
Ward’s Minimum-Variance Method	Distance between two clusters is ANOVA sum of squares between the two clusters added up over all the variables. At each generation, the within-cluster sum of squares is minimized over all partitions obtainable by merging two clusters from the previous generation.	Strongly biased toward producing clusters with roughly same number of observations. Also very sensitive to outliers