Risk Budgeting

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