Case 066Portfolio construction and optimisationHard

Market weights of 70% large caps and 30% mid caps imply mid caps beat large caps by 1% a year. An analyst's view says 3%, held with the same confidence as the market prior. What blended spread results, and which way do the weights move?

The situation

Chandrabhaga Capital runs an Indian equity book split between large caps and mid caps at market weights, 70% and 30%. Large caps have volatility of 15%, mid caps 18%, and the correlation between them is 0.85. Working backwards from the market weights, the returns the market implies have mid caps beating large caps by 1% a year.

The mid cap analyst argues that earnings growth and a narrowing valuation gap will make the spread 3% a year. The CIO says to hold the view with the same confidence as the market prior, no more and no less, and to use the Black-Litterman approach to combine them.

Your task

What expected spread do you use after blending, which way and roughly how far do the weights move, and why not simply plug in the analyst's 3%?

Quick check

With the view held at the same confidence as the market prior, what blended spread do you use?

Worked solution

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

30-second answerThe answer to give first

The blended spread is 2%, and weights move towards mid caps: from 30% to about 65% on these inputs. Equal confidence means the market's 1% and the analyst's 3% pull equally, so the estimate lands halfway. Plugging in the raw 3% would push an optimiser to about 100% mid caps, a corner. The blend's tilt carries about 3.3% tracking error; a 2% cap would hold mid caps near 51%.

Step 1Where does the market's 1% come from?

From the weights themselves. If investors in aggregate hold 70% large caps and 30% mid caps, there is a set of expected returns that makes those weights the sensible choice, given the risks. Working backwards from the weights, the volatilities and the correlation gives the market's implied spread, here 1% a year; this reverse optimisationSolving for the expected returns that would make the current market weights optimal, instead of choosing weights from guessed returns. is the starting point Black-Litterman uses. On these inputs it implies large caps at about 7.2% and mid caps at 8.2% above cash.

Step 2Why does equal confidence land exactly halfway?

Think of two friends estimating the drive time to Pune: one says four hours, the other five, and you trust them equally. You plan for four and a half. Black-Litterman blends the prior and the view in proportion to the confidence in each, so equal confidence gives equal weight and the spread becomes 2%. If the CIO trusted the view only a quarter as much as the prior, the blend would move a fifth of the way, to 1.4%. The weight on the view is its confidence divided by the total confidence.

The market's implied 1% and the analyst's 3% carry equal confidence, so the blended mid cap spread lands halfway at 2%; with a quarter of the confidence in the view it would reach only 1.4%.

Step 3Which way do the weights move, and how far?

Towards mid caps, and further than many people expect. The weights move by the change in the spread divided by risk aversion times the spread's variance. The mid-minus-large spread has a volatility of only 9.5%, because the two move closely together, so an extra 1% of expected spread shifts about 35 points of weight: mid caps go from 30% to about 65%. Plug the analyst's 3% straight in and the shift doubles, to about 100% mid caps and nothing in large caps, the kind of corner answer that makes optimisers distrusted.

Starting from 30% mid caps, the blended 2% estimate moves the book to about 65% mid caps, a 2% tracking error cap holds it near 51%, and trusting the 3% view fully sends it to about 100%.

The relationship

\mu_{\text{blend}} = \frac{c_p\,\pi + c_v\,Q}{c_p + c_v} = \frac{1 \times 1\% + 1 \times 3\%}{2} = 2\%

pi	the market's implied spread, 1%
Q	the analyst's view, 3%
c_p, c_v	confidence in the prior and in the view, equal here

What it says in wordsThe blended estimate is a confidence-weighted average of what the market implies and what the analyst believes.

Step 4What would you actually do with the book?

Use the blend, then apply the risk budget. A 35 point tilt in a spread with 9.5% volatility carries about 3.3% of tracking error, so if the book's budget is 2%, scale the tilt to about 21 points and hold mid caps near 51%. Then write down what would make the analyst wrong: mid cap earnings downgrades, or the valuation gap widening further. The limitation is that confidence is itself a judgement; Black-Litterman makes it explicit and consistent, but it cannot tell you how right the analyst is.

Where candidates lose it

The common loss is feeding the analyst's 3% straight into an optimiser. The optimiser treats 3% as certain and piles into mid caps, and the book ends up with a concentrated bet no one intended.

The second is answering 2% and stopping. The interviewer then asks what happens to the weights, and the surprise is how far a one point change moves them when the two assets are highly correlated.

What the interviewer asks next

The CIO doubles her confidence in the view. What blended spread and what weights result?
Why does high correlation between the two assets make the weights so sensitive?
How would you express a view that mid caps return 14% in absolute terms, rather than relative to large caps?
What would you use for the risk aversion coefficient, and why does it matter?

Company names and figures are illustrative.