Inputs/assumptionsMatrix mathKey portfolio calculations (marginal VaR, Inc'l VaR, Component VaR)

Portfolio inputs: A B ConfidencePosition $100.00 $100.00 Critical zVolatility 10.00% 14.00%Tot Portfolio $200.00 Matrix math:Correlation 0.20

Positions (x')Var-Covar 0.01 0.0028 100 100Matrix: 0.0028 0.0196

PortfolioVariance (matrix) $352.00 Positions (x')Variance (alt) $352.00 100 100Volatility $18.76Volatility (alt.) $18.76VaR $43.65

Individual Positions A BPositions $100.00 $100.00Volatility 10.00% 14.00%(sigma)(X) 1.28 2.24Individual VaR $23.26 $32.57Beta 0.727 1.273Marginal VaR 0.159 0.278Marginal VaR 0.159 0.278

Incremental Position A B+/- Position $0.010 $0.000(x Marg'l VaR) 0.00159 0.0000

Incremetal VaR 0.00159 0.00000

Component VaR $15.87 $27.77 $43.65Percent Contrib. 36% 64%

http://www.bionicturtle.com/forum/viewthread/599/

99.00%2.33

Var-Covar Matrix (sigma) Positions (x)0.01 0.0028 $100.00

0.0028 0.0196 $100.00

(sigma)(x) x'(sigma)x1.28 $352.002.24

1.28 1.282.24 2.24

Question:Assume a $200 two-asset portfolio with equal positions in both assets ($100 + $100). Asset #1 has volatility of 10%, Asset #2 has volatility of 14%. Their correlation is 20%. Our desired confidence is 99%.(i) What is portfolio volatility?(ii) What is portfolio VaR and diversified VaR and what is the difference?(iii) What are the individual VaRs?(iv) What is the incremental VaR?(v) What are the component VaRs and percentage contributions?(vi) If correlation is perfect (1.0), how will portfolio VaR compare to individual VaRs?

Answer:

(i) Kudos if you can use matrix math to derive (see XLS). But also,Variance = ($100^2)(10%^2)+($100^2)(14%^2)+(2)($100)($100)(0.20)(10%)(14%) = 352Volatility = SQRT(352) = $18.8 Or, you can use percentage weights instead of dollar positions:Volatility = SQRT[(50%^2)(10%^2)+(50%^2)(14%^2)+(2)(50%)(50%)(0.20)(10%)(14%)] = 9.38%(9.38%)($200) = $18.8

(ii)No difference!With 99%, the normal deviate = NORMSINV(99%) = 2.33Diversified VaR = (2.33)($18.8) = $43.6This is a relative VaR. No expected returns are given and expected return is not netted, which would be an absolute VaR.

(iii)($100)(10%)(2.33) = $23.26, and($100)(14%)(2.33) = $32.57

(iv) The incremental VaR can be approximated with: marginal VaR * trade.Assuming the marginal VaR = 0.159 (see next), and the trade is +$10, thenapproximate incremental VaR = (10)(0.159) = $1.59But please note that using the marginal VaR is only an approximation. The true incremental VaR is given by the difference between Portfolio VaR (before trade) - Portfolio VaR (after trade)

(v)First, we want the marginal VaR. Note in XLS we can get this two ways.But, given that the betas are, respectively, 0.727 and 1.273, Marginal VaR = Portfolio VaR/Portfolio Value * beta. In this case,Asset #1 Marginal VaR = $43.6/$200 * 0.727 beta = 0.159Asset #2 Marginal VaR = $43.6/$200 * 1.273 beta = 0.278Component VaR = Position * Marginal VaR. In this case,Asset #1 Component VaR = (0.159)($100) = $15.87 Asset #2 Component VaR = (0.278)($100) = $27.77Percentage contributions are:Asset #1 Component VaR % = $15.87/$43.6 = 36%Asset #2 Component VaR = $27.77/43.6 = 64%

(vi)If correlation = 1.0, sum of individual VaRs will EQUAL portfolio VaR.But for imperfect correlation (rho < 1), sum of individual VaRs will be GREATER THAN portfolio VaR.Sum of component VaRs, by definition, will be EQUAL to portfolio VaR.

Question:Assume a $200 two-asset portfolio with equal positions in both assets ($100 + $100). Asset #1 has volatility of 10%, Asset #2 has volatility of 14%. Their correlation is 20%. Our desired confidence is 99%.(i) What is portfolio volatility?(ii) What is portfolio VaR and diversified VaR and what is the difference?(iii) What are the individual VaRs?(iv) What is the incremental VaR?(v) What are the component VaRs and percentage contributions?(vi) If correlation is perfect (1.0), how will portfolio VaR compare to individual VaRs?

Answer:

(i) Kudos if you can use matrix math to derive (see XLS). But also,Variance = ($100^2)(10%^2)+($100^2)(14%^2)+(2)($100)($100)(0.20)(10%)(14%) = 352Volatility = SQRT(352) = $18.8 Or, you can use percentage weights instead of dollar positions:Volatility = SQRT[(50%^2)(10%^2)+(50%^2)(14%^2)+(2)(50%)(50%)(0.20)(10%)(14%)] = 9.38%(9.38%)($200) = $18.8

(ii)No difference!With 99%, the normal deviate = NORMSINV(99%) = 2.33Diversified VaR = (2.33)($18.8) = $43.6This is a relative VaR. No expected returns are given and expected return is not netted, which would be an absolute VaR.

(iii)($100)(10%)(2.33) = $23.26, and($100)(14%)(2.33) = $32.57

(iv) The incremental VaR can be approximated with: marginal VaR * trade.Assuming the marginal VaR = 0.159 (see next), and the trade is +$10, thenapproximate incremental VaR = (10)(0.159) = $1.59But please note that using the marginal VaR is only an approximation. The true incremental VaR is given by the difference between Portfolio VaR (before trade) - Portfolio VaR (after trade)

(v)First, we want the marginal VaR. Note in XLS we can get this two ways.But, given that the betas are, respectively, 0.727 and 1.273, Marginal VaR = Portfolio VaR/Portfolio Value * beta. In this case,Asset #1 Marginal VaR = $43.6/$200 * 0.727 beta = 0.159Asset #2 Marginal VaR = $43.6/$200 * 1.273 beta = 0.278Component VaR = Position * Marginal VaR. In this case,Asset #1 Component VaR = (0.159)($100) = $15.87 Asset #2 Component VaR = (0.278)($100) = $27.77Percentage contributions are:Asset #1 Component VaR % = $15.87/$43.6 = 36%Asset #2 Component VaR = $27.77/43.6 = 64%

(vi)If correlation = 1.0, sum of individual VaRs will EQUAL portfolio VaR.But for imperfect correlation (rho < 1), sum of individual VaRs will be GREATER THAN portfolio VaR.Sum of component VaRs, by definition, will be EQUAL to portfolio VaR.

Question:Assume a $200 two-asset portfolio with equal positions in both assets ($100 + $100). Asset #1 has volatility of 10%, Asset #2 has volatility of 14%. Their correlation is 20%. Our desired confidence is 99%.(i) What is portfolio volatility?(ii) What is portfolio VaR and diversified VaR and what is the difference?(iii) What are the individual VaRs?(iv) What is the incremental VaR?(v) What are the component VaRs and percentage contributions?(vi) If correlation is perfect (1.0), how will portfolio VaR compare to individual VaRs?

Answer:

(i) Kudos if you can use matrix math to derive (see XLS). But also,Variance = ($100^2)(10%^2)+($100^2)(14%^2)+(2)($100)($100)(0.20)(10%)(14%) = 352Volatility = SQRT(352) = $18.8 Or, you can use percentage weights instead of dollar positions:Volatility = SQRT[(50%^2)(10%^2)+(50%^2)(14%^2)+(2)(50%)(50%)(0.20)(10%)(14%)] = 9.38%(9.38%)($200) = $18.8

(ii)No difference!With 99%, the normal deviate = NORMSINV(99%) = 2.33Diversified VaR = (2.33)($18.8) = $43.6This is a relative VaR. No expected returns are given and expected return is not netted, which would be an absolute VaR.

(iii)($100)(10%)(2.33) = $23.26, and($100)(14%)(2.33) = $32.57

(iv) The incremental VaR can be approximated with: marginal VaR * trade.Assuming the marginal VaR = 0.159 (see next), and the trade is +$10, thenapproximate incremental VaR = (10)(0.159) = $1.59But please note that using the marginal VaR is only an approximation. The true incremental VaR is given by the difference between Portfolio VaR (before trade) - Portfolio VaR (after trade)

(v)First, we want the marginal VaR. Note in XLS we can get this two ways.But, given that the betas are, respectively, 0.727 and 1.273, Marginal VaR = Portfolio VaR/Portfolio Value * beta. In this case,Asset #1 Marginal VaR = $43.6/$200 * 0.727 beta = 0.159Asset #2 Marginal VaR = $43.6/$200 * 1.273 beta = 0.278Component VaR = Position * Marginal VaR. In this case,Asset #1 Component VaR = (0.159)($100) = $15.87 Asset #2 Component VaR = (0.278)($100) = $27.77Percentage contributions are:Asset #1 Component VaR % = $15.87/$43.6 = 36%Asset #2 Component VaR = $27.77/43.6 = 64%

(vi)If correlation = 1.0, sum of individual VaRs will EQUAL portfolio VaR.But for imperfect correlation (rho < 1), sum of individual VaRs will be GREATER THAN portfolio VaR.Sum of component VaRs, by definition, will be EQUAL to portfolio VaR.

Question:Assume a $200 two-asset portfolio with equal positions in both assets ($100 + $100). Asset #1 has volatility of 10%, Asset #2 has volatility of 14%. Their correlation is 20%. Our desired confidence is 99%.(i) What is portfolio volatility?(ii) What is portfolio VaR and diversified VaR and what is the difference?(iii) What are the individual VaRs?(iv) What is the incremental VaR?(v) What are the component VaRs and percentage contributions?(vi) If correlation is perfect (1.0), how will portfolio VaR compare to individual VaRs?

Answer:

(i) Kudos if you can use matrix math to derive (see XLS). But also,Variance = ($100^2)(10%^2)+($100^2)(14%^2)+(2)($100)($100)(0.20)(10%)(14%) = 352Volatility = SQRT(352) = $18.8 Or, you can use percentage weights instead of dollar positions:Volatility = SQRT[(50%^2)(10%^2)+(50%^2)(14%^2)+(2)(50%)(50%)(0.20)(10%)(14%)] = 9.38%(9.38%)($200) = $18.8

(ii)No difference!With 99%, the normal deviate = NORMSINV(99%) = 2.33Diversified VaR = (2.33)($18.8) = $43.6This is a relative VaR. No expected returns are given and expected return is not netted, which would be an absolute VaR.

(iii)($100)(10%)(2.33) = $23.26, and($100)(14%)(2.33) = $32.57

(iv) The incremental VaR can be approximated with: marginal VaR * trade.Assuming the marginal VaR = 0.159 (see next), and the trade is +$10, thenapproximate incremental VaR = (10)(0.159) = $1.59But please note that using the marginal VaR is only an approximation. The true incremental VaR is given by the difference between Portfolio VaR (before trade) - Portfolio VaR (after trade)

(v)First, we want the marginal VaR. Note in XLS we can get this two ways.But, given that the betas are, respectively, 0.727 and 1.273, Marginal VaR = Portfolio VaR/Portfolio Value * beta. In this case,Asset #1 Marginal VaR = $43.6/$200 * 0.727 beta = 0.159Asset #2 Marginal VaR = $43.6/$200 * 1.273 beta = 0.278Component VaR = Position * Marginal VaR. In this case,Asset #1 Component VaR = (0.159)($100) = $15.87 Asset #2 Component VaR = (0.278)($100) = $27.77Percentage contributions are:Asset #1 Component VaR % = $15.87/$43.6 = 36%Asset #2 Component VaR = $27.77/43.6 = 64%

(vi)If correlation = 1.0, sum of individual VaRs will EQUAL portfolio VaR.But for imperfect correlation (rho < 1), sum of individual VaRs will be GREATER THAN portfolio VaR.Sum of component VaRs, by definition, will be EQUAL to portfolio VaR.

A B C D Port0.61% 0.20% 0.24% 0.64% 0.42%0.02% 0.58% -0.30% 0.40% 0.18%1.00% 1.45% 1.15% 2.15% 1.44%1.55% 1.73% 1.76% 0.99% 1.51%1.94% 1.87% 1.70% 0.89% 1.60%1.85% 0.58% 0.54% 1.53% 1.12%0.13% 0.52% 0.46% 0.30% 0.35%0.82% 0.55% 0.53% 0.04% 0.48%0.75% 1.30% 0.95% 1.13% 1.03%2.05% 0.63% 1.15% 1.27% 1.27%0.60% 0.76% 0.53% 0.60% 0.62%0.82% 1.12% 0.33% 1.04% 0.83%

-3.70% -1.72% -1.67% -1.78% -2.22%2.17% 2.22% 1.76% 3.09% 2.31%1.24% 2.37% 1.91% 2.38% 1.98%

-0.83% 0.17% -0.55% 1.02% -0.05%-1.00% 0.15% 0.41% 0.91% 0.12%-1.11% -1.85% -1.69% -0.26% -1.23%1.57% 1.45% 1.44% 2.76% 1.80%

-1.66% -0.81% -1.84% -1.07% -1.35%-0.29% -0.10% 0.24% 0.26% 0.03%0.77% 0.88% 1.28% 1.74% 1.17%0.54% 0.32% 0.73% 0.93% 0.63%

-1.43% -2.26% -3.08% -1.66% -2.11%-0.83% -0.73% -1.89% 0.12% -0.83%-5.75% -2.44% -2.25% -3.18% -3.41%-5.61% -5.37% -4.22% -1.48% -4.17%-1.44% -1.88% -1.58% 1.85% -0.76%-0.06% 0.63% 0.26% -0.14% 0.17%2.21% 0.00% -0.37% 2.43% 1.07%0.48% -1.08% -0.79% 0.70% -0.17%

-0.52% 0.19% -0.26% -0.04% -0.15%1.62% 0.14% 0.51% -0.30% 0.49%2.27% 1.37% 1.80% 2.10% 1.88%0.31% 0.28% -0.38% 0.35% 0.14%1.54% 1.30% 1.16% 0.28% 1.07%

Mean 0.07% 0.13% 0.00% 0.61% 0.20%Std Dev 1.94% 1.54% 1.47% 1.34% 1.46%Skew -1.60 -1.48 -0.97 -0.60 -1.27Kurt 2.86 3.39 0.67 0.82 1.73

VarCovarMat0.000365 0.000246 0.000233 0.000198 0.0003650.000246 0.000231 0.000208 0.000140 0.0002460.000233 0.000208 0.000210 0.000138 0.0002330.000198 0.000140 0.000138 0.000175 0.000198

CorrelMat1.00000 0.84911 0.83976 0.78304 1.000000.84911 1.00000 0.94580 0.69716 0.849110.83976 0.94580 1.00000 0.72100 0.839760.78304 0.69716 0.72100 1.00000 0.78304

TotalWeights 25% 25% 25% 25% 100%Position 100 100 100 100 400

Sample PopulationPort Mean 0.20%Port Vol 1.46% 1.44%Port Vol 5.8324 1 5.7508 5.8324Variance 34.017 33.07208Port VaR ( Raw ) 13.568 13.37844 13.568Port VaR ( Rel ) -3.1900%

Vol (Pop) 1.91% 1.52% 1.45% 1.32% 1.91%Vol (Sample ) 1.94% 1.54% 1.47% 1.34% 1.94%Sigma (x) 0.104134 0.082543 0.078939 0.065104 1

CL 99.00%Z-Score 2.33CF Mod 0.73

Individual VaR 4.5057 3.5838 3.4211 3.1205Slope 1.26 1.00 0.95 0.79Beta 1.26 1.00 0.95 0.79Marginal VaR 0.0427 0.0339 0.0324 0.0267 0.13568Marginal VaR 0.0427 0.0339 0.0324 0.0267 0.13568

Component VaR 4.2722 3.3864 3.2386 2.6710 13.56831% 25% 24% 20% 1

New Weights 33% 33% 33% 0% 100%Increment 133.333 133.333 133.333 0 400Incremetal VaR 5.6963 4.5153 4.3181 0.0000 14.530

-3.4304%

Modified0.000246 0.000233 0.000198 0.00069 0.00039 0.00041 0.000340.000231 0.000208 0.00014 0.00039 0.00049 0.00034 0.000220.000208 0.00021 0.000138 0.00041 0.00034 0.00031 0.00027

0.00014 0.000138 0.000175 0.00034 0.00022 0.00027 0.00026

Modified0.84911 0.83976 0.78304 1.0000 0.6749 0.8796 0.81411.00000 0.94580 0.69716 0.6749 1.0000 0.8778 0.62820.94580 1.00000 0.72100 0.8796 0.8778 1.0000 0.95190.69716 0.72100 1.00000 0.8141 0.6282 0.9519 1.0000

Sample Population

1.92% 1.89%7.6695 7.562258.821 57.187

17.84182 17.84182 17.592-4.2584% -4.2584%

1.52% 1.45% 1.32% 2.63% 2.21% 1.77% 1.60%1.54% 1.47% 1.34% 2.67% 2.24% 1.80% 1.63%

1 1 1 0.184042 0.144549 0.133815 0.10946

0.8819 1.0544 0.5160 0.49606.2138 5.2082 4.1799 3.78576.2138 5.2082 4.1799 3.7857

Modified Beta 1.29 1.01 0.94 0.770.0574 0.0451 0.0417 0.03420.0574 0.0451 0.0417 0.0342

5.7420 4.5098 4.1749 3.415132% 25% 23% 19%

33% 33% 33% 0%133.333 133.333 133.333 0.000

7.6560 6.0131 5.5666 0.0000

0.1784180.178418

17.84182

100%400

19.2357-4.6069%