Amath 546/Econ 589 Risk Budgeting and Risk...

Amath 546/Econ 589

Risk Budgeting and Risk Reporting

Eric Zivot

April 10, 2013

Risk decomposition using Euler’s theorem

Let RM(w) denote a portfolio risk measure that is a homogenous function ofdegree one in the portfolio weight vector w Euler’s theorem gives the additiverisk decomposition

RM(w) = 1RM(w)

1+ 2

RM(w)

2+ · · ·+

RM(w)

=X=1

RM(w)

= w0RM(w)

w

Terminology:

MCR =RM(w)

= marginal contribution to risk of asset i,

CR = ·MCR = contribution to risk of asset i,

PCR =CRRM(w)

= percent contribution of asset i

Then

RM(w) = 1 ·MCR1 +2 ·MCR2 + · · ·+ ·MCR= CR1 + CR2 + · · ·+ CR

1 =CR1RM(w)

+ · · ·+ CRRM(w)

= PCR1 + · · ·+ PCR

Risk Decomposition for Portfolio SD

RM(w) = (w) = (w0Σw)12 = w0

(w)

wNow

(w)

w=

(w0Σw)12

w=1

2(w0Σw)−122Σw

=Σw

(w0Σw)12=

Σw

(w)

⇒ (w)

= ith row of

Σx

(w)

Remark: In R, the PerformanceAnalytics function StdDev() performs this de-composition based on the sample covariance matrix of returns.

Example: 2 asset portfolio

(w) = (w0Σw)12 =³21

21 + 22

22 + 21212

´12Σw =

Ã21 1212 22

!Ã12

!=

Ã1

21 +212

222 +112

!Σw

(w)=

⎛⎝ ³1

21 +212

´(w)³

222 +112

´(w)

⎞⎠

Then

MCR1 =³1

21 +212

´(w)

MCR2 =³2

22 +112

´(w)

CR1 = 1 ×³1

21 +212

´(w) =

³21

21 +1212

´(w)

CR2 = 2 ×³2

22 +22

´(w) =

³22

22 + 1212

´(w)

and

PCR1 = CR1(w) =³21

21 +1212

´2(w)

PCR2 = CR2(w) =³22

22 +1212

´2(w)

How to Interpret and Use MCR

MCR =(w)

≈ ∆

∆⇒ ∆ ≈ MCR ·∆

However, in a portfolio of assets

1 + 2 + · · ·+ = 1

so that increasing or decreasing means that we have to decrease or increaseour allocation to one or more other assets. Hence, the formula

∆ ≈ MCR ·∆

ignores this re-allocation effect.

If the increase in allocation to asset is offset by a decrease in allocation toasset then

∆ = −∆

and the change in portfolio volatility is approximately

∆ ≈ MCR ·∆ +MCR ·∆

= MCR ·∆ −MCR ·∆

=³MCR −MCR

´·∆

1 2 21 22 1 2 12 120.175 0.055 0.067 0.013 0.258 0.115 -0.004875 -0.164

Table 1: Example data for two asset portfolio.

Consider two portfolios:

• equal weighted portfolio 1 = 2 = 05

• long-short portfolio 1 = 15 and 2 = −05

MCR CR PCR = 01323

Asset 1 0.258 0.5 0.23310 0.11655 0.8807Asset 2 0.115 0.5 0.03158 0.01579 0.1193

= 04005Asset 1 0.258 1.5 0.25540 0.38310 0.95663Asset 2 0.115 -0.5 -0.03474 0.01737 0.04337

Table 2: Risk decomposition using portfolio standard deviation.

Interpretation: For equally weighted portfolio, increasing 1 from 05 to 06decreases 2 from 0.5 to 0.4. Then

∆ ≈ (MCR1 −MCR2) ·∆

= (023310− 003158)(01)= 002015

For the long-short portfolio, increasing 1 from 15 to 16 decreases 2 from-0.5 to -0.6. Then

∆ ≈ (MCR1 −MCR2) ·∆

= [025540− (−003474)] (01)= 002901

Remark

The reallocation does not have to be all in one asset. Suppose, the reallocationis spread across all of the other assets 6= so that

∆ = −∆ s.t.X 6=

= 1

Then X 6=

∆ = −X 6=

∆ = −∆

X 6=

= −∆

and

∆ ≈ MCR ·∆ +X 6=

MCR ·∆

=

⎛⎝MCR ·−X 6=

·MCR

⎞⎠∆

In matrix notation the result can be written as

∆ ≈³MCR0α

´∆

MCR = (MCR1 MCR)0

α = (−1 −−1 1−+1 − )0

Interpreting MCR when (w) = (w) or (w)

• ∆ = −∆

∆ (w) =³MCR

−MCR

´·∆

∆(w) =³MCR −MCR

´·∆

• ∆ = −∆ s.t.P 6= = 1

∆ (w) ≈³MCR 0α

´∆

∆(w) ≈³MCR 0α

´∆

Beta as a Measure of Asset Contribution to Portfolio Volatility

For a portfolio of assets with return

(w) = 11 + · · ·+ = w0R

we derived the portfolio volatility decomposition

(w) = 1(w)

1+ 2

(w)

2+ · · ·+

(w)

= w0

(w)

w(w)

w=

Σw

(w)(w)

= ith row of

Σw

(w)

With a little bit of algebra we can derive an alternative expression for

MCR =(w)

= ith row of

Σw

(w)

Definition: The beta of asset with respect to the portfolio is defined as

=((w))

((w))=

((w))

2(w)

Result: measures asset contribution to (w) :

MCR =(w)

= (w)

CR = (w)

PCR =

Remarks

• By construction, the beta of the portfolio is 1

=((w) (w))

((w))=

((w))

((w))= 1

• When = 1

MCR = (w)

CR = (w)

PCR =

• When 1

MCR (w)

CR (w)

PCR

• When 1

MCR (w)

CR (w)

PCR

MCR CR PCR = 01323

Asset 1 0.258 0.5 1.76 0.23310 0.11655 0.8807Asset 2 0.115 0.5 0.24 0.03158 0.01579 0.1193

= 04005Asset 1 0.258 1.5 0.64 0.25540 0.38310 0.95663Asset 2 0.115 -0.5 -0.09 -0.03474 0.01737 0.04337


Remarks:

• For the equally weighted portfolio, Asset 1 is a risk enhancer and Asset 2is a risk reducer

• For the long-short portfolio, 2 0!

− − Decomposition of Portfolio Volatility

Recall,

MCR =(w)

= ith row of

Σw

(w)=

((w))

(w)

Using

= ((w)) =((w))

(w)⇒ ((w)) = (w)

gives

MCR =(w)

(w)=

Then

CR = ×MCR = × ×

= allocation× standalone risk× correlation with portfolio

Remarks:

• × = standalone contribution to risk (ignores correlation effects withother assets)

• CR = × only when = 1

• If 6= 1 then CR ×

MCR CR PCR = 01323

Asset 1 0.258 0.5 0.90 0.23310 0.11655 0.8807Asset 2 0.115 0.5 0.27 0.03158 0.01579 0.1193

= 04005Asset 1 0.258 1.5 0.99 0.25540 0.38310 0.95663Asset 2 0.115 -0.5 -0.30 -0.03474 0.01737 0.04337


Remarks:

• For the equally weighted portfolio, both assets are positively correlatedwith the portfolio

• For the long-short portfolio, Asset 2 is negatively correlated with the port-folio

Remark:

• Goldberg, Hayes, Menchero and Mitra (2009) “Extreme Risk Manage-ment”, Barra Technical Report, extends the and − − decompo-sitions to ES

Asset $ Asset VaR MCVaR CVaR PCVaRAsset 1 10 .10 .01 .10 -.03 .003 .01 .10Asset 2 20 .20 .02 .12 -.04 .002 .02 .11... ... ... ... ... ... ... ... ...Asset N 5 .05 .01 .07 -.07 .010 .04 .13Portfolio 100 1 .03 .08 .08 1

Table 5: Portfolio VaR Report

Portfolio VaR and ES Reports

A common portfolio risk report summarizes asset and portfolio risk measuresas well as risk budgets

Amath 546/Econ 589 Risk Budgeting and Risk...

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