Portfolio
optimisation via
Polynomial Goal
Programming
Overview
 Asset allocation via Polynomial Goal Programming (PGP)
 Application to ASX50
 Performance metrics
 Limitations and extensions
Asset allocation via PGP
 Previous approaches can only optimise one objective function subject
to constraints.
 Maximise Sharpe ratio subject to ESG constraints (efficient frontier)
 Minimise tracking error subject to ESG constraints
 Minimise CO2 subject to tracking error constraints
 Polynomial goal programming (PGP) is a numerical technique used to
optimise multiple objective functions subject to constraints.
 Can use solver (or any language that has an optimisation package)
Asset allocation via PGP
 We will illustrate via an asset allocation that optimises over the first 2
moments (mean/variance) and CO2 i.e
 Maximise returns
 Minimise variance
 Minimise CO2/$1M USD in revenue.
 PGP is implemented in 2 steps
 Step 1: Optimise each objective separately to determine the aspired
levels -
 Step 2: Optimise a joint objective function that seeks to minimise
deviations from each of the aspired levels.
2
2,, ,p p pCO   
PGP – Step 1: Obtain aspired levels
Covariance matrix1
1) Return optimisation
Max ( )
Subject to 1
0
T
p p
n
i
i
i
E R W
w
w
 

 



 2 2
1
2) Variance optimisation
Min ( )
Subject to 1
0
T
p p p
n
i
i
i
E R W W
w
w
 

   




The solution for each optimisation = aspired level.
This represents the best possible outcome for the joint
optimisation in step 2.
2
2, 2
1
2 2
3) optimisation
Min CO
Subject to 1
0
where CO is 1vector of / $1Min USD revenues for each asset
T
p
n
i
i
i
CO
W CO
w
w
n CO







PGP-Step 2: Minimise deviations from aspired levels
denotes investor preferences e.g
(1,1,1) all goals equal
(1,1,0) mean variance optimisation
(1.5,1,1) pursue excess returns
(1,1.2,0.8) more emphasis on risk, less on CO2
Because mean can be <0 take absolute values
Denominators in objective function standardise each
deviation by the aspired level from step 1
2
, i.e what is achieved < aspiration given additional constraints
, i.e what is achieved > aspiration given additional constraints
T
p
T
p
W
W W
 


 


1 2 3, ,  
Solutions for
each objective
2 2, i.e what is achieved>aspiration, given additional constraints
T
pW CO CO 
1 2 3
31 2
2
2,
1
2
2
2 3 2,
1
Min 1 1 1
Subject to
1
0
0
p p p
T
p
T
p
T
p
n
i
i
i
i
dd dZ
CO
W d
W W d
W CO d CO
w
w
d
  
 
 


     
 
  
 




 



We add 1 to each standardised deviation so that increases in lambda
increase Z (this is not the case if the absolute value of the
standardised deviation is <1).
PGP Example: ASX50 stocks
 We apply the approach to our 50-asset portfolio of ASX50 stocks.
 The calculations are in the spreadsheets “PGP_step1.xlsx” and
“PGP_step2.xlsx”
 Note: solver is an iterative algorithm, so starting values may make a
difference.
 I would suggest re-setting the weights to equally weighted if your solution is not
the same as mine.
 For your assignment you should also make sure your results are robust to changes
in the starting weight vector.
Spreadsheet: same as before
Assume we are at Jun 9, 2021 and seek to optimise for the next day i.e Jun 10, 2021
Like before we i) estimate the expected returns and cov matrix using the last 252 trading days
ii) impose zero weights on the stocks without CO2 reporting (highlighted in red in spreadsheet)
PGP-Step 1 (mean maximisation)
Solving for the mean target = 0.256
Input result into the goal cell BG4
Note green weights vector reveals
allocation is exclusively in ATP
stock
PGP- Step 1 (variance minimisation)
Solving for the variance target = 0.578
Input result into the goal cell BG5
Note: green weights vector has more assets with non
zero weights. Less likely to load onto 1 asset as the
portfolio variance is also a function of covariances
PGP – Step 1 (CO2 minimisation)
Solving for the CO2 target = 0.940
Input result into the goal cell BG5
Note: green weights vector is 100% in
ATP stock (again)
PGP-Example Step 1
The goals from step 1 have
now been set in the
spreadsheet.
Ready to do step 2
For step 2 we set d1,d2,d3
to arbitrary small values
and all lambdas to unity.
PGP: Step 2 (minimise deviations from target)
Optimisation is more stable if
CO2 is on a similar scale to returns
and variance. Therefore divide CO2
by 100 to get target of 0.0094
PGP: Step 2 (minimise deviations from target)
Final
solution
1
2
2
2
2 2,
outcome < aspiration (given additionalconstraints)
0.122 0.134 0.256
outcome > aspiration (given additionalconstraints)
1.812 1.234 0.578
outcome > aspirat
T
p
T
p
T
p
T
p
T
p
W
W d
W W
W W d
W CO CO
 
 



    
 
     






2 3 2,
ion (given additional constraints)
0.020 0.011 0.009T pW CO d CO    
Differences in starting values & investor preferences
Need to consider different starting values for the weights and d’s.
Without dividing CO2 by 100, we can see some sensitivity in the results
If different solutions are found for a given set of lambdas, you pick the solution with the lowest Z.
Change lambdas to consider different investor preferences
(1,1,1) – base case: Sharpe ratio 0.067, only invest in 6/50 assets
As reduce lambda on CO2 (from 0.75 to 0.1), the Sharpe ratio increases, CO2 increases and number of stocks held increase.
We can also consider changing the lambdas for returns and variance.
Performance metrics
 Performance should be considered out of sample (OOS).
 Weight vectors from PGP should be used to determine the allocation over
a given horizon, say the next month.
 At the end of the month, the estimation window should be rolled forward
and the optimisation re-performed. This will determine the allocation for
the next month etc….
 Returns on their own do not consider risk
 Sharpe ratio may not suitable (only considers mean and variance)
 Consider downside risk metrics
Performance metrics: downside risk
 Sortino ratio: improves Sharpe ratio by dividing excess returns by a semi-variance
 ௣ is the portfolio return, ௙ the risk-free rate and ௣ the semi-variance (based on negative
returns)
 Semi-variance = variance for a normal/symmetric distribution.
 Maximum drawdown (MDD): measures the largest price drop from peak to trough
 $600 not used as it’s not a new high
 Expected shortfall (ES): Average returns that are worse than the Value at Risk.
 If VaR is at 95% level, expected shortfall is the average of the worst 5% of portfolio returns
.
௣ − ௙
௣
Limitations and extensions
 Could consider other objectives as well, e.g also minimise water
usage or maximise the E or S score.
 More targets from step 1 may mean the algorithm has difficulty finding an
optimal solution.
 PGP is numerical and there may be local maxima.
 Need to check step 2 solution with a variety of starting values for the
weight vector (equal weighting is always a good start) and d1,d2,d3 etc...
 Important as the number of assets, targets and constraints increase.
 PGP cannot be directly tied to a utility function. Other approaches
 consider higher moments via Taylor series expansions of utility
 derive optimal weights for given distributions (analytical)
 numerically optimise portfolio using tail risk (like VaR)