Constraining Portfolio Turnover in Bond Allocation Optimization

# Portfolio Turnover Constraint


# Portfolio Turnover Constraint












I have a few bonds and OAS and Duration for each. I had a Linear programming type of problem where I had to maximize OAS and keep duration <= constraint. There are few other constraints. I could easily model them using `linprog` in MATLAB.

Unfortunately the portfolio turnover is too high. I want to put a constraint on this turnover. Something like 10% of total portfolio. Since it is not a linear programming problem anymore I am a bit stuck. Any thoughts will be greatly helpful?

## Answer by Brian B (score 3, accepted)

https://quant.stackexchange.com/a/14932

At each rebalancing day, you were previously maximizing

$$ \vec{w}^* \vec{r} -\lambda \vec{w}^* \Sigma \vec{w} $$

Now, you need to combine this with a way of expressing your trading constraint mathematically. Let's say your previous weights were $\vec{p}$. Then your 10% constraint translates to specifying that

$$ 0.1 \geq \sum |w_i-p_i| $$

This can be handled by `fmincon`, or perhaps even by some of Matlab's simpler optimizers.

## Answer by ash (score -1)

https://quant.stackexchange.com/a/14941

Thanks all for the adivse. I tried `fmincon` . Unfortunately it was not converging to any solution and I was not getting an idea of which constraint is breaking it and which direction I should modify the constrain to reach a solution. So this is what I finally did.

- Continue using `linprog` for optimization.

- Calculate $w_b$ and $w_s$ and calculate the portfolio churn as $w_c = w_b+w_s$

- if $w_c > 0.10*w_{total}$

- Create a vector of rates duration such that $\delta_r \in [\frac{\delta_{rTot}}{10},\delta_{rTot}] $

- Create a vector of spread duration such that $\delta_s \in [\frac{\delta_{sTot}}{10},\delta_{sTot}] $

- For each $\delta(i)_r$ I draw spreads from $\delta(1)_s...\delta(10)_s$ and re-run `linprog` . I save each successful solutions $\delta(i)_s$,$\delta_r(j)$,OAS and $w_c$.

Hence in the end I have all the feasible solutions and corrosponding boundries of durations. Now I can suggest a few optimial portfolios with acceptable durations , OAS and transaction costs. (I have a function which models this for me based on Outstanding Amount , buy/sell Amounts etc) It is brute force but works for me. This wasy I not just know the infeasibles but also all posible feasibles under an investment strategy be it

- Increade OAS and keep spread and duration under a boundry

- Keep OAS same but reduce durations

- A combination of the two with low transaction cost

Just wanted to let other know in case it provide any help to anyone.

EDIT

I ended up coding up using fmincon . I modelled the obective to increase the OAS and non linear constrain bit as

```
function [c,ceq]=NonLiniearConstraints(~,x,w_i)
    churn = 0.10;
   res = sum(abs(x-w_i));
   c(1) = res - 2*churn; % Buy + Sell = Total churn 
   ceq = [];
end
```

I verified that it produces same results as that of `isqlin` which tries to minimize churn using least sqaures meathod. So I have marked Brian's answer as the correct one.