Risk Contributions
Matrix notation is used throughout this chapter. Matrix operations are very convenient when dealing with data on portfolio / benchmark segment level. Not every body is aware that Microsoft Excel can execute matrix operations.
Definition of Absolute and Relative Risks
Absolute Benchmark Risk: The risk of the benchmark portfolio is described by the variance or standard deviation of the expected return of this portfolio. Mathematically, the absolute risk is estimated by taking the benchmark or target weights and multiplying them through a variance-covariance matrix...
var(b) = w'(b) * V * w(b)
var(b)... benchmark variance
w(b)... benchmark (target) weights
w'(b)... transposed benchmark (target) weights
V... variance-covariance matrix of asset classes
Absolute Portfolio Risk: The risk of the actual portfolio defined and calculated the same way...
var(p) = w'(p) * V * w(p)
var(p)... portfolio variance
w(p)... actual asset class weights
V... variance-covariance matrix of asset classes
Relative Portfolio Risk: The active portfolio consist of the active positions defined as differences in asset class weights between the portfolio and the benchmark. The relative portfolio risk is then defined as...
var(a) = w'(a) * V * w(a)
with
w(a) = w(p) - w(b)
var(a)... relative portfolio risk (=risk of the active positions)
w(a)... active weights
V... variance-covariance matrix of asset classes
The active weights w(a) can be negative or positive. The sum of the active weights equals zero.
Tracking Error is the square root of relative portfolio risk, which can be expressed as...
TE = w'(a)*V*w(a)/sqrt[w'(a)*V*w(a)]
Marginal Risk Contributions
The marginal risk contribution can be thought of as the rate of change in risk with respect to a small percentage change in the size o the position.
Mathematically speaking, the marginal contribution to total risk from an individual bet is the first derivative of the risk measure vis-à-vis the individual bet.
The marginal contribution of the bet in the ith asset class (zi) to tracking error is therefore...
MC(TE, i) = w(a,i) *∂(TE)/ ∂w(a,i)
= w(a,i)*{{w'(a)*V}/sqrt[w'(a)*V*w(a)]}
MC(TE, i)... Marginal risk contribution of bet i to TE
w(a,i)... active bet i
TE... tracking error
V... variance-covariance matrix
The same calculation can be done for absolute portfolio and benchmark risks...
MC(var(p), i) = w(p,i) *∂(var(p))/ ∂w(p,i)
= w(p,i)*{{w'(p)*V}/sqrt[w'(p)*V*w(p)]}
MC(var(p), i)... Marginal risk contr. of bet i to abs. portfolio risk
w(p,i)... active bet i
var(p)... absolute portfolio risk
V... variance-covariance matrix
MC(var(b), i) = w(b,i) *∂(var(b))/ ∂w(b,i)
= w(b,i)*{{w'(b)*V}/sqrt[w'(b)*V*w(b)]}
MC(var(b), i)... Marginal risk contr. of bet i to abs. portfolio risk
w(b,i)... active bet i
var(b)... absolute portfolio risk
V... variance-covariance matrix
The risk contributions can also be expressed as percent since the marginal contribution add up to total risk...
MC%(TE, i) = MC(TE, i) / TE
MC%(var(p), i) = MC(TE, i) / var(p)
MC%(var(b), i) = MC(TE, i) / var(b)
Worked Example
You can download a worked example of a risk contribution here.
Funding Assumption
Underlying the concept of marginal risk contribution is an assumption about how an increase in a weight is funded: marginal contributions implicitly assume that the additional investments are funded out of cash (riskfree bank account). Note that other funding assumption are probably more realistic (for example, funding one increase in weight with one decrease in another weight, or funding one increase in a weight with the an equally distributed decrease in all other weights).
Alternative Formulas for Marginal Risk Contributions
As not everybody is familiar with matrix notation, below the formulas used in this section expressed as sums...
std(b) = sum[var(i)*wb(i)^2, i] + sum[sum[wb(i)*wb(j)*cov(i,j), j <>i], i]
std(p) = sum[var(i)*wp(i)^2, i] + sum[sum[wp(i)*wp(j)*cov(i,j), j <>i], i]
std(a)= sum[var(i)*wa(i)^2, i] + sum[sum[wa(i)*wa(j)*cov(i,j), j <>i], i]
MC(std(a),i) = b(i)*std(a) = wa(i)*var(i) + sum[wa(j)*cov(i,j), j<>i]
MC(std(b),i) = b(i)*std(b) = wb(i)*var(i) + sum[wb(j)*cov(i,j), j<>i]
MC(std(p),i) = b(i)*std(p) = wb(i)*var(i) + sum[wp(j)*cov(i,j), j<>i]
To see the above formulas in action, you can download an Excel spreadsheet here.
Incremental Risk Contributions
...
Component Risk Contributions
...
|
