Case 100Portfolio constructionCore

Vindhavan's two assets have equilibrium expected returns of 6% and 7%. A manager believes A will beat B by 3% and holds that view with 50% confidence. In a simple two-asset Black-Litterman setting, how do the blended expected returns move, and what happens to the weights?

The situation

Vindhavan Capital allocates between two invented assets. Asset A has volatility 20% and asset B 15%, with correlation 0.5. Backing out expected returns from the market's holdings gives equilibrium returns of 6% for A and 7% for B, which correspond to market weights of 13.6% in A and 86.4% in B.

A portfolio manager has a view: A will beat B by 3 percentage points over the next year. Asked how sure she is, she says 50%. The committee wants to know what expected returns to use, and what the view does to the weights.

Your task

Blend the equilibrium returns with the view at 50% confidence, show how the result moves with confidence, and say what the blended returns do to the weights.

Quick check

Equilibrium says B beats A by 1 point; the view says A beats B by 3 points. At 50% confidence, what is the blended spread A minus B?

Worked solution

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

30-second answerThe answer to give first

At 50% confidence the blended returns are 7.54% for A and 6.54% for B: the spread moves from -1% halfway to the +3% view, to +1%. A moves +1.54 points and B -0.46, because A is the more volatile asset and the covariance spreads the view unevenly. Full confidence would put A at 9.08% and B at 6.08%; zero leaves the equilibrium. The weights move from 13.6% in A to 32.5% at half confidence and 51.4% at full.

Step 1What does the view say, and what does the equilibrium say?

Two friends guess the weight of a parcel. One has a scale that is usually right and says 7 kg; the other has lifted it and says 10. If you half trust the lifter, you say 8.5, not 10 and not 7. Black-Litterman is that averaging done properly: the equilibrium returnsThe expected returns implied by the market portfolio and the covariance matrix, found by reverse optimisation. They are the returns at which the market as it stands is optimal. are the scale, the manager's view is the lifter, and confidence sets the weight on each. Here the equilibrium says A earns 6% and B 7%, a spread of -1%. The view says the spread is +3%. The gap between the two claims is 4 points, and that gap is what confidence will scale.

Step 2How does 50% confidence become numbers?

The view is a statement about one combination, A minus B, whose variance under the covariance matrix is 0.0325, a standard deviation of 18.0%. Setting the view's own uncertainty equal to that variance, which is what 50% confidence means in this convention, puts half the weight on the view, so the blended spread is -1% + 0.5 x 4 = +1%; the covariance then shares the move between the assets, A rising 1.54 points to 7.54% and B falling 0.46 points to 6.54%. The split is uneven because A is the more volatile asset and the view is a relative one: a claim that A beats B is more likely to be about A moving than about B, in the ratio of their covariances with the spread, 3.3 to 1 here. The scalar tau that scales the prior covariance cancels in this two-asset case with one view, which is why the answer needs only the confidence.

The relationship
μBL=π+ΣP⊤ cPΣP⊤ (Q−Pπ)PΣP⊤=0.04+0.0225−2(0.5)(0.2)(0.15)=0.0325,Q−Pπ=0.04\mu_{BL} = \pi + \Sigma P^{\top}\,\frac{c}{P \Sigma P^{\top}}\,(Q - P\pi) \qquad P\Sigma P^{\top} = 0.04 + 0.0225 - 2(0.5)(0.2)(0.15) = 0.0325, \quad Q - P\pi = 0.04
piequilibrium expected returns, 6% and 7%
P, Qthe view: P = (1, -1) picks A minus B, Q = 3% is the claimed value
cconfidence, here 0.5, which replaces tau and omega when there is one view
What it says in wordsStart from equilibrium, measure how far the view disagrees, scale that disagreement by confidence, and spread it across the assets in proportion to how each one moves with the viewed combination.
Half the confidence moves each forecast half the way to the view5%6%7%8%9%10%A: equilibrium 6%blend 7.54%full view 9.08%B: equilibrium 7%blend 6.54%full view 6.08%-1%+0%+1%+3%50% confidence: spread +1.0%0% confidence100%blended A minus BA moves +1.54 points and B -0.46: the more volatile asset carries more of the view.
Vindhavan's equilibrium returns of 6% and 7% move to 7.54% and 6.54% at 50% confidence, halfway along the arrows to the full-view values of 9.08% and 6.08%, and the blended spread rises in a straight line from -1% to +3% as confidence goes from zero to one.
Step 3What does the blend do to the weights?

The reason to blend rather than plug the view in is what the optimiser does next. Fed the raw view, a mean-variance optimiser treats +3% as certain and swings the allocation; fed the blend, it moves the weight in A from 13.6% to 32.5% at half confidence and to 51.4% at full, a measured path instead of a jump. The table traces the path. Two properties are worth saying aloud: at zero confidence the weights are the market weights, which is the model's anchor, and because the view is relative, the weights still sum to one and the total risk budget barely changes; only the tilt does. The limitation: the 50% is a convention, not a measurement. Different conventions for turning confidence into the view's variance give different paths, so state yours, and never let a view's confidence be set by the person who holds the view without a track record behind it.

Confidence in the viewE[A]E[B]A minus BWeights A / B
0%6.00%7.00%-1.00%13.6% / 86.4%
25%6.77%6.77%+0.00%23.1% / 76.9%
50%7.54%6.54%+1.00%32.5% / 67.5%
75%8.31%6.31%+2.00%42.0% / 58.0%
100%9.08%6.08%+3.00%51.4% / 48.6%
Vindhavan's blended returns and unconstrained weights as confidence in the view rises from zero to one; the spread moves linearly from -1% to +3%, and the weight in A climbs from the market's 13.6% to 51.4%.

Where candidates lose it

The common loss is taking half the view, +1.5%, as the blended spread. Half confidence means half the distance from the equilibrium spread of -1% to the view of +3%, which is +1%; the equilibrium has an opinion too.

The second is moving A alone. A relative view shifts both assets, in proportion to their covariance with the viewed combination, so B's expected return falls even though the view said nothing about B on its own.

What the interviewer asks next

  • How would the answer change if A and B had equal volatility, and why would the move then split evenly?
  • What does tau do when there are two views with different confidences, and why did it cancel here?
  • How would you set the confidence from the manager's past record of calls rather than from her own estimate?
  • If the view were absolute, A returns 9%, rather than relative, how would B's blended return change?

Company names and figures are illustrative.