Case 009Rebalancing, implementation and costsCore

A Rs 500 crore hybrid fund targets 65/35 and has drifted to 69/31. Trading costs 30 basis points each way. Compare a full rebalance with trading back only to the 67/33 band edge, in cost and in risk.

Man GroupBoston · 2022

The situation

Anuradha Hybrid Fund has Rs 500 crore and a strategic mix of 65% equity and 35% bonds, with a rebalancing band of 2 points either side. After a strong equity run it sits at 69/31. Dealing costs, spread plus market impact, are about 30 basis points on each trade, buying or selling.

For the risk estimate, assume equity volatility of 18%, bond volatility of 5% and a correlation of 0.1 between them.

Your task

What does each option cost, how much risk does each remove, and which would you do?

Quick check

Trading to the band edge costs half as much as a full rebalance. How much of the risk does it remove?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Trade to the band edge: it costs Rs 6 lakh against Rs 12 lakh and removes 75% of the tracking risk. A full rebalance sells Rs 20 crore of equity and buys Rs 20 crore of bonds; the edge trade moves Rs 10 crore each way. Tracking error against the 65/35 target falls from 0.73% to 0.36%, and because risk grows with the square of the gap, the first half of the trade does most of the work.

Step 1Why is there a band at all?

Because every trade costs money and a small drift costs very little risk. A band says: tolerate small drift for free, and trade only when drift becomes expensive enough in risk to be worth paying for. A thermostat works the same way. It does not fire the heater every time the room cools by a tenth of a degree; it waits for a gap worth closing, because switching on and off has a cost of its own.

Step 2What does each trade cost?

Going back to 65% means selling 4% of Rs 500 crore, Rs 20 crore of equity, and buying Rs 20 crore of bonds. At 30 basis points on each side, that is Rs 12 lakh; stopping at 67% trades half as much and costs Rs 6 lakh. Costs scale in a straight line with the size of the trade, which is the key to the comparison.

Step 3How much risk does each remove?

Measure risk as tracking errorThe volatility of the difference between the fund and its target mix, here the 65/35 strategic portfolio. against the 65/35 target. Being 4 points overweight equity and 4 underweight bonds is a spread bet whose volatility is 18.2% on each point, so 0.73% today, 0.36% at 67% and zero at 65%. The harm from drift grows with the square of the gap, so halving the gap removes 75% of the tracking variance. Costs are a straight line; risk is a curve. That asymmetry is why trading to the edge wins.

The relationship
TE=∣w−0.65∣σe2+σb2−2ρσeσb=0.04×18.2%=0.73%\text{TE} = |w - 0.65| \sqrt{\sigma_e^2 + \sigma_b^2 - 2\rho\sigma_e\sigma_b} = 0.04 \times 18.2\% = 0.73\%
wequity weight, 69% today
\sigma_e, \sigma_bequity and bond volatility, 18% and 5%
\rhotheir correlation, 0.1
What it says in wordsTracking error is the size of the overweight times the volatility of equities against bonds.
Two ways back from 69%: all the way to target, or to the band edgeno-trade band 63% to 67%60%61%63%67%69%71%72%target 65%today 69%Full: trade Rs 20 crore each waycost Rs 12 lakh, tracking error 0.73% to 0Edge: trade Rs 10 crore each waycost Rs 6 lakh, tracking error 0.73% to 0.36%half the cost buys75% of the risk cutRisk measured as tracking variance against the 65/35 target
From 69%, a full rebalance costs Rs 12 lakh and removes all of the 0.73% tracking error, while trading to the 67% band edge costs Rs 6 lakh and removes 75% of the tracking variance.
Equity weightTrade each way, Rs croreCost, Rs lakhFund volatilityTracking errorExtra loss if equities fall 30%, Rs crore
69%, do nothing0012.67%0.73%6.0
67%, band edge10612.33%0.36%3.0
65%, full rebalance201212.00%0.00%0.0
Trading to 67% costs Rs 6 lakh and halves the extra crash loss from Rs 6 crore to Rs 3 crore; the last Rs 6 lakh buys the smallest part of the risk reduction.
Step 4Which would you do, and what would you do before trading at all?

Trade to the edge. The portfolio-choice literature has shown since Davis and Norman, 1990, that with costs proportional to trade size the efficient policy is a no-trade band, rebalancing to its edge rather than its centre. Before paying anything, use the free rebalancers: send new inflows to bonds and pay redemptions from equity, which may close the gap at no dealing cost. The limit to say out loud is that the edge rule assumes drift is random; if the committee believes equities are expensive, that is a tactical view, and it should be argued and recorded as one, not hidden inside a rebalancing rule.

Where candidates lose it

Most candidates compare only costs and pick the cheaper trade without measuring what the risk reduction is worth, or compare only risk and rebalance fully every time. The question is the trade-off, and the square in the risk term is what settles it.

The second miss is forgetting that costs apply on both legs. Selling equity and buying bonds each cost 30 basis points, so a Rs 20 crore rebalance costs Rs 12 lakh, not Rs 6 lakh.

What the interviewer asks next

  • The fund expects Rs 15 crore of inflows next month. How does that change what you trade today?
  • Why might a fund use a wider band for an illiquid asset class?
  • How would you set the band width itself, knowing costs and volatilities?

Asked at Man Group, Investment Management, Boston, 2022 (Wall Street Oasis): How do you understand portfolio risk and transaction cost?

Company names and figures are illustrative.