# 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 12 w

⎞

w0Σw and =

Example: Portfolio risk decomposition for 2 risky asset portfolio = 11 + 22 2 2 2 = 1 2 + 2 2 + 212 12 1 2

=

³

2 2 1 2 + 2 2 + 212 12 1 2

´12

To get an additive decomposition for 2 write

2 2 2 = 1 2 + 2 2 + 212 12 1 2

=

Here we can split the covariance contribution 21212 to portfolio variance evenly between the two assets and deﬁne 2 1 2 + 1212 = 1 2 2 2 + 1212 = 2

³

2 2 1 2 + 1212 + 2 2 + 1212 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 + 1212 2 2 + 1212 1 2 = + 2 2 + 1 1 1 2 12 = sd contribution of asset 1 2 2 2 + 12 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) = 21 + 2 2 = 2(1 + 2 ) 6= · (1 2)

Let (1 2) =

q

2 2 1 + 2 Then

( · 1 · 2) =

q

q 22 + 22 = (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)12 = () Then

( · w) = ( · w)0( · w)

³

´12

= · w0 w

³

´12

= · (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)12 = (w) = (w VaR(w) = −1−(w) × 0 1− = 1−(w) = (w) + (w) × 1−

() = −0 ⎝(w)...

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