Risk Optimization tools

Portfolio Manager's Wish:-

        Every portfolio manager wishes to construct optimum portfolios that are less risky, profitable & with less investment. This is going to be a uphill task, In this post we have demonstrated Portfolio Risk Optimization processes & tools.

Optimize Portfolio Risk How to? Process

Portfolio Manager can optimize[Reduce Risk, Improve Returns] portfolio by doing:-

               1) Conscious allocation of economic capital.
               2) Trade off risk and return.


Optimize Portfolio Risk How to? Tools are:-

	Tool	Measures	Helps in deciding
	Incremental  VaR	Effective Portfolio Risk, after adding a new instrument.	Whether to add a new instrument to portfolio?
	Marginal      VaR	Effective Portfolio Risk, after increasing unit investment in existing instrument.	Whether to invest more in existing instrument?
	Component VaR	Effective Portfolio Risk, decomposed between instruments.	Whether to retain/drop an instrument?

Optimization Tools: In Detail

Marginal VaR

Measures change in portfolio VaR resulting from adding additional dollar to an instrument(ith).     

Marginal VaR of ith component is  ∆VaRi  =   α σip/σp  =(VaR/P * Bi)

Component VaR

Measures VaR contribution of every instrument to Portfolio VaR.

Component VaR of ith component is  ∆VaRi  =  (VaR * Bi * wi)

Incremental VaR

Measures VaR contribution of a new instrument to Portfolio VaR.

Incremental VaR of NewInstrumenta = VaRp+a - VaRp


Optimization Tools: Implementation

Following Steps are followed:-

         1) Calculate Instrument Statistics[Return,StdDev]
         2) Calculate Portfolio    Statistics[Weighing,Return,StdDev]
         3) Calculate Related      Statistics[Betas].
         4) Calculate VaR           [Marginal VaR, Componenet VaR, Incremental VaR]

Instrument Return:-

InstrumentsValue(past value)
	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10
01-Jan-2001	2.3	2	3	23.3	12	13	3	23.3	12	13
02-Jan-2001	1.4	5.1	2.6	1.8	32.1	21.6	2.6	1.8	32.1	21
03-Jan-2001	21.6	2.1	3.3	4.1	2.2	3.4	11.5	2.2	12.4	1.5
04-Jan-2001	21.1	12	13	21.1	22	31	21.1	22	31	19
05-Jan-2001	2.15	2.12	3.21	12.51	2.21	3.12	2.15	2.21	13.1	2.34
06-Jan-2001	11.95	21.22	13.11	11.95	12.22	31.11	12.95	21.21	3.21	1.23

                                      From the historical returns obtain returns for every date.
                                       

InstrReturns
	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10
01-Jan-2001	.	.	.	.	.	.	.	.	.	.
02-Jan-2001	-0.3913	1.55	-0.13333	-0.92274	1.675	0.661538	-0.13333	-0.92274	1.675	0.61538
03-Jan-2001	14.428	-0.58823	0.269230	1.277777	-0.93146	-0.84259	3.423076	0.222222	-0.61370	-0.9285
04-Jan-2001	-0.023	4.714285	2.939393	4.1463415	9	8.117647	0.834782	9	1.5	11.6666
05-Jan-2001	-0.898	-0.82333	-0.753077	-0.407109	-0.899545	-0.89935	-0.89810	-0.89954	-0.57741	-0.8768
06-Jan-2001	4.55813	9.00943	3.084112	-0.044764	4.529411	8.971153	5.023255	8.597285	-0.75496	-0.4743

            do i = 1 to 10;

                 CurrValue     = InstrumentsHistory [,i];

                 PreValue       = Lag(InstrumentsHistory [,i],1);

                 Returns         = ((CurrValue-PreValue)/PreValue);

                 InstrReturns  = InstrReturns||Returns;

           end;

                 Portfolio Returns:-

PortfolioWeight
	W1	W2	W3	W4	W5	W6	W7	W8	W9	W10
P1	0	0.1	0.2	0.4	0.3	0	0	0	0	0
P2	0.2	0	0	0.4	0.2	0.1	0	0	0	0.1
P3	0	0	0.2	0.4	0.3	0	0.1	0	0	0
P4	0.3	0.1	0.2	0.4	0	0	0	0	0	0

                         PortfolioReturn= Return(PortfolioValue=(PortfolioWeight*Instrument))

                                                     

PortReturns
	P1	P2	P3	P4
01-Jan-2001	.	.	.	.
02-Jan-2001	-0.170554	-0.209743	-0.194645	-0.79926
03-Jan-2001	-0.721441	-0.410103	-0.630728	3.1428571
04-Jan-2001	4.9432177	2.2017417	3.8053528	1.0656285
05-Jan-2001	-0.653875	-0.708885	-0.669671	-0.649812
06-Jan-2001	1.0226959	1.0006229	0.8950031	1.0158388

                  do i = 1 to 4;
                       PortfolioValue    = (PortfolioWeight[i,]#InstrumentsHistory)[,+];
                       PrePortfolioValue = Lag(PortfolioValue,1);
                      CurrPortReturns   = ((PortfolioValue-PrePortfolioValue)/PrePortfolioValue);
                      PortReturns       = PortReturns||CurrPortReturns  ;
                 end;
                               
                              PortStdev=Sqrt(Var(PortfolioReturn))

PortStdev
P1	P2	P3	P4
2.3743983	1.2105273	1.8782371	1.6012189

Portfolio Beta Calculations:-

                              PortfolioBeta= Corr p,i * (InstrStdDev/PortStdev)

PortBeta
	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10
P1	-0.272492	1.3931548	0.4549135	0.2571137	1.7647082	1.6598638	0.2099907	1.2973967	0.103451	2.0616687
P2	1.6034421	3.0741911	1.1897248	0.8405295	3.1152606	3.6627218	1.0297197	2.9689203	0.1014574	3.5945565
P3	1.0334214	1.9813219	0.7667798	0.5417228	2.0077912	2.360631	0.6636563	1.9134746	0.0653895	2.3166983
P4	3.2325549	0.2582333	0.5621477	1.1437981	-0.261683	0.3076701	0.9341662	2.2445145	-0.383511	-1.019062

Measure of  relationship between Portfolio loss & Instrument loss.


Let's Calculate VaR:-

MVar:- 

PortMVar
	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10
P1	.	0.2507888	0.0818913	0.0462843	0.317674	.	.	.	.	.
P2	0.1510749	.	.	0.079194	0.2935171	0.3450984	.	.	.	0.338676
P3	.	.	0.1164923	0.0823007	0.3050318	.	0.1008254	.	.	.
P4	0.3948454	0.0315423	0.0686644	0.139711	.	.	.	.	.	.

            For Portfolio P1 investing in I4 is less risky.

ComponentVar:-

PortCVar
	I1	I2	I3	I4	I5	I6	I7	I8	I9	I10
P1	.	0.3307904	0.2160292	0.2441962	1.257036	.	.	.	.	.
P2	0.3882021	.	.	0.4069936	0.7542216	0.4433825	.	.	.	0.4351309
P3	.	.	0.2880389	0.4069936	1.1313324	.	0.1246504	.	.	.
P4	1.5528084	0.0413488	0.1800243	0.7325885	.	.	.	.	.	.

            For Portfolio P1 more risk contribution is done by I5 and less contribution is done by I3.

Incremental VaR:-

          Full Revaluation should be used to calculate this (Varp+a - Vara)

Conclustion:-

Using these three VaR measures portfolio manager can decide on optimizing portfolio by reducing risk. But this is only one end of the coin :). Follow this space to do it more effectively.