# Risk Budgeting

Topics: Value at risk, Investment, Variance Pages: 10 (1357 words) Published: May 6, 2013
AMATH 546/ECON 589 Risk Budgeting

Eric Zivot

April 10, 2012

Outline

• Portfolio Calculations • Risk Budgeting • Reverse Optimization and Implied Returns

Portfolio Risk Budgeting

• Additively decompose (slice and dice) portfolio risk measures into asset contributions

• Allow portfolio manager to know sources of asset risk for allocation and hedging purposes

• Allow risk manager to evaluate portfolio from asset risk perspective

Portfolio Calculations Let 1      denote simple returns on  assets, and let 1      denote P portfolio weights such that   = 1 =1 Portfolio return:

R = (1      ) w = (1     )0 1 = (1     1)0  = w0R =
=1  X

 w01 = 1

Portfolio mean and variance: Let R be a random vector with [R] = μ = (1     )0 (R) = [(R − μ)(R − μ)0] = Then  = w0μ  2 =  ⎛

Σ

⎜ ⎜ =⎜ ⎜ ⎝

2 12 · · · 1 1 ⎟  12  2 · · · 2 ⎟ 2 . . . ⎟ ... . . . ⎟ ⎠ 1  2 · · · 2  ³ ´ 0Σw 12 w

w0Σw and   =

Example: Portfolio risk decomposition for 2 risky asset portfolio  = 11 + 22 2 2  2 = 1 2 + 2 2 + 212 12  1 2

 =

³

2 2 1 2 + 2  2 + 212 12 1 2

´12

To get an additive decomposition for 2 write 
2 2  2 = 1 2 + 2 2 + 212 12  1 2

=

Here we can split the covariance contribution 21212 to portfolio variance evenly between the two assets and deﬁne 2 1 2 + 1212 = 1 2 2 2 + 1212 = 2

³

2 2 1 2 + 1212 + 2 2 + 1212  1 2

´

³

´

variance contribution of asset 1 variance contribution of asset 2

We can also deﬁne an additive decomposition for 
2 2 1 2 + 1212 2 2 + 1212 1 2  = +   2 2 +    1 1 1 2 12 = sd contribution of asset 1  2 2  2 + 12 12 2 = sd contribution of asset 2 

Euler’s Theorem and Risk Decompositions

• When we used  to measure portfolio risk, we were able to easily derive an additive risk decomposition.

• If we measure portfolio risk by VaR or ES it is not so obvious how to deﬁne individual asset risk contributions.

• For portfolio risk measures that are homogenous functions of degree one in the portfolio weights, Euler’s theorem provides a general method for additively decomposing risk into asset speciﬁc contributions.

Homogenous functions and Euler’s theorem First we deﬁne a homogenous function of degree one. Deﬁnition 1 homogenous function of degree one Let  (1     ) be a continuous and diﬀerentiable function of the variables 1       is homogeneous of degree one if for any constant   0  ( · 1      · ) =  ·  (1     ) Note: In matrix notation we have  (1     ) =  () where = (1     )0 Then  is homogeneous of degree one if  (·) = · ()

Examples Let  (1 2) = 1 + 2 Then  ( · 1  · 2) =  · 1 +  · 2 =  · (1 + 2) =  ·  (1 2) 2 2 Let  (1 2) = 1 + 2  Then 2 2 2 2  ( · 1  · 2) = 21 + 2 2 = 2(1 + 2 ) 6=  ·  (1 2)

Let  (1 2) =

q

2 2 1 + 2 Then

 ( · 1  · 2) =

q

q 22 + 22 =  (2 + 2) =  ·  (   )  1 1 2 2 1 2

Repeat examples using matrix notation Deﬁne = (1 2)0 and 1 = (1 1)0 Let  (1 2) = 1 + 2 =01 = f () Then  ( · w) = ( · w)0 1 =  · (w01) =  ·  (w)

2 2 Let  (1 2) = 1 + 2 =0=  () Then

 ( · w) = ( · w)0( · w) = 2 · w0w 6=  ·  (w) Let  (1 2) = q
2 2 1 + 2 = (0)12 =  () Then

 ( · w) = ( · w)0( · w)

³

´12

=  · w0 w

³

´12

=  ·  (w)

Consider a portfolio of  assets = (1     )0 with initial value 0 and let  ∈ (0 1) denote a conﬁdence level

R = (1     )0 w = (1     )0 [R] = μ cov(R) = Σ R ∼ (μ Σ) Deﬁne  = (w) = w0R  = (w) = w0μ  2 = 2(w) = w0Σw   0Σw)12   =  (w) = (w  VaR(w) = −1−(w) × 0    1− = 1−(w) = (w) +  (w) × 1−

() = −0 ⎝(w)...