Quality Money Management : Process Engineering and Best Practices for Systematic Trading and Investment

248 CHAPTER ◆ 2 7 Define Performance Controls

where:

● Q1 is the benchmark return, the weighted sum of the benchmark sector weights and

benchmark sector returns.

● Q2 is the weighted sum of portfolio weights and benchmark returns.

● Q3 is the benchmark sector weights times the portfolio sector returns. Q3 shows the

return that would have been realized if the asset allocation stuck to the benchmark

weights, but stock selection was active.

● Q4 is the portfolio return.

In terms of quadrant returns, the interaction term provides the missing piece that makes

the attributes sum exactly to the active return:^3

Asset allocation  Q2  Q1

Security selection  Q3  Q1

Interaction  Q4  Q3  Q2  Q1

Total value-added  Q4  Q1

Since the attributes sum to the value added, the Brinson model is additive.

The Brinson–Fachler model decouples the sector returns relative to or in relationship

with the overall return. If a sector had positive return, but a return nevertheless below the

overall return, then the asset allocation difference will be negative and the sector selec-

tion effect negative. In such a case, the portfolio manager would be penalized for invest-

ing too much money in a bad sector, a sector that lowered the index benchmark return. If,

however, the sector outperformed the overall benchmark, then the sector effect would be

positive. The Brinson–Fachler model looks at how the sector did in comparison with the

benchmark and reports the results.^4

27.3. LOOP 3: Benchmark VaR Calculations and Software

Monte Carlo simulation approximates the behavior of instrument prices by using com puter-

generated random numbers to build hypothetical price paths through time. The VaR can be

read directly from the distribution of simulated portfolio values. A main advantage of the

Monte Carlo method is that it is able to cover a wide range of possible values in financial

variables and fully account for correlations. Essentially, the method has five steps:

● Specify a stochastic process for each financial variable as well as process param-

eters. Parameters, such as standard deviation and correlations, may be based upon

historical data or implied volatilities. A commonly used stochastic process model

for random price movement is geometric Brownian motion, where the change in the

asset price, S , is a function of the mean return,  , the standard deviation of returns,

 , the change in time, t , and a standard normal random number,  , where:

SStt 1 ()t t

The Cox, Ingersoll, and Ross model is a commonly used stochastic, mean-reverting

process of interest rates where:

drttt()r dt r dz