Sharpe Weights

One question that I have always had is how do you construct a portfolio without an optimizer and a large number of predictions about your investment universe. The reason I have this question is because of a word I see from time to time when active portfolio management is discussed, the word “conviction weights”. A typical active stock picking portfolio managers will not spend time forecasting a full correlation matrix, but that doesn’t mean either that the portfolio he will build will be unreasonable from a risk return perspective. So the question is what can we do to have weights that are still optimal from an ex ante sharpe ratio perspective but in a very simplified setup and without an optimizer?

If you’re a quant and you’re reading this you will probably tell me I am reinventing hot water, and probably you won’t be far off the mark. This is not research, more a way to put together my personal thoughts on this, not something I am doing for my work either. I did do some limited simulations, but nothing like you’d find in a proper paper.

I make the following assumptions in terms of distributions of returns and risks:

• The Returns of the assets are normally distributed
• The Sharpe Ratios of assets are normally distributed with zero mean and variance one (Yes that’s not really correct.)
• The asset’s volatilities as a cross section are lognormally distributed.

Step 1: Maximally shrinking the correlation matrix or in plain English, using just one correlation number for the whole investment universe.

This makes sense for two reasons:

-We know that one of the best ways to not get extreme weights out of an optimizer is to shrink the correlation matrix (Ledoit and Wolf, 2003)

-In the extreme scenario where there is only one correlation between all asset pairs, the optimal portfolio weights have a closed form (See the Elton / Gruber textbook p 203)

Even after this simplification, you will, in theory still need an optimizer because you will almost certainly want to add at minimum two types of constraints: No short sales and a concentration threshold.

Step 2: Choosing a concentration constraint

DeMiguel & al propose the use of a 1-norm or 2-norm constraint on weights for portfolio optimization defined as:

?for p=1,2

The 2-norm constraint limits the distance of the portfolio weights to the equal weight passive portfolio thereby having the same effect as shrinking the covariance matrix towards identity and making expected returns more equal. In other words the constraint has a similar effect to tweaking how the manager’s views impact the weights in a Bayesian setup such as Black-Litterman.

The 2-norm is also a constraint on the sum of the square of weights ?and which is equivalent to having a constraint on the Herfindahl-Hirschman Index (HHI):

It turns out the HHI Index is a very good tool for us because we can formulate an approximation of k, the number of equal sized bet in a portfolio with just one correlation, and assets with an average volatility?and a variance of v around that average (The formula below is a simplification of the one in Choueifaty and al, Properties of the most diversified portfolio).

The constraint on the HHI index is therefore equivalent to a constraint on the number of independent bets in the portfolio.

In our case we will not use directly the HHI. First we?will note that the inverse of the HHI, HHI^-1 represents the effective number of bets in the portfolio when the correlation is zero, then we will adopt as a measure of concentration the following ratio:

Which represents the effective number of bets in the portfolio when correlation is zero divided by N, the number of assets of assets in the portfolio.

Step 3: Simplifying the investor views

• His objective is to maximize the ex-ante Sharpe Ratio of his portfolio.
• The investor has selected assets which he believes are in the top x% of the Sharpe Ratios of his investment universe. We will vary the selectivity of this investor by varying x between 10% and 40%. The investor is also able to RANK these assets by preference, even if he doesn’t have an opinion about their precise risk and rewards.
• The number of securities in the portfolio is between 50 and 100 in our simulations. We chose this number somewhat arbitrarily but we believe it covers typical portfolios.

Step 4: We will now set weights directly using the investor’s views and explicitly formulate the concentration of the portfolio.

Before we check whether it is possible in practice, with the assumptions we have made, to dispense with the use of an optimizer, let us look at how we could use Sharpe Ratios as weights, and in particular how we can satisfy both the requirement that the weights sum to 1 and as well incorporate the constraint on the HHI index we mentioned earlier.

Sharpe Ratio S is a normally distributed random variable:

The investor only selects Assets whose Sharpe Ratio exceed a threshold K. K is for example the minimum Sharpe Ratio for which all higher Sharpe Ratios are in the top 25%.

S|S>K is therefore a truncated normal random variable:

and

Where

?is the probability density function of the standard normal distribution and

?is its cumulative distribution function.

Let us define:??

We introduce constant K’ which will be subtracted from S, with ?(this ensures the weights are all positive) and will later help us adjust weights to get a desired level of concentration in the portfolio:

???and ???

As we have N assets in the portfolio, we need to ensure that the average weight is 1/N so that the weights sum to 1:

Considering weight W as a random variable, we want to find an expression such that has expected Value E(W)=1/N, which is:

We now have an explicit formula to compute the weight in the portfolio of an asset whose Sharpe Ratio exceeds K. Can we compute the “Effective N%” concentration for this portfolio?

The HHI Index in terms of the random variable W is:

?so ?

And

We end up with

And finally,

So assuming the Sharpe Ratios of the portfolio are normally distributed and selected above threshold K we can now chooses a level of concentration for the portfolio by computing the concentration when we change K’<K and then obtain explicitly a weight.

o???Maximum attainable portfolio concentration with this method

The Maximum concentration is attained when K’=K. Here is what the maximum levels are depending on the selectivity of the investor:

If you had a 50 name portfolio, using an Effective N% of 60% would mean the weights are equivalent to holding a 50*0.6 = 30 name equal weight portfolio. We think few Investors would want to run a much lower number of names since if they wanted less diversification it would probably make sense to just run a very small number of names with an equal weight.

We note as well that such levels of maximum concentration always enable the investor to be in line with UCITs rules by construction. In the maximum selectivity where the investor selects Sharpe Ratios in the top 10%, the approximative maximum weight is 7.6% (so less than 10%), and the total value of positions exceeding a 5% weight is 19.3% of AuM (so less than 40%). In effect when selecting the top 10% of Sharpe Ratios, the 10% maximum weight rule and the 40% maximum in the weights above 5% rule are both respected as long as the total number of assets is equal to or higher than 39.

Let us note, though, that this method does NOT give you a lot of flexibility on the concentration levels.

We now recapitulate the steps to get from the views to the weights:

In the general case , when you have selected the assets and ranked them from the least attractive (k=1) to the most attractive (k=N):

• You have selected the names in the top x of Sharpe ratio where x is a percentage, ie x=40%=0.4 for example.
• You obtain K as

• You chose K’<K?in order to target a particular concentration:

• Each asset k has percentile ?

• The Sharpe Ratio of Asset k is

• The weight of asset k is ?

A practical Example in Excel

?

Cells B1 to B5 are our Inputs.

Cells B5’s formula is: =NORM.S.DIST(B2,FALSE)/(1-NORM.S.DIST(B2,TRUE)) and the weight Multiplier in B6 is =1/(B4*(B5-B3)) . Finally the Explicit calculation of the portfolio concentration in B7 is =((B5-B3)^2)/(1+B3^2+B5*(B2-2*B3))

At this stage you need the ranking of the assets by attractivity:

?

Assume this is a 50 name portfolio. In this case we have selected the top 40%

?

The least attractive asset is above the 60th percentile by definition. We will now assign a percentile to each asset. From the least to the most attractive (k=1 to 50) we assign the percentile p=0.6+0.5*0.4/50 to the first asset and p=0.6+(k-0.5)*0.4/50 to the kth asset. The Sharpe Ratio of each asset is simply the inverse cumulative normal distribution function, so cell F3 above is simply?NORM.S.INV(E3). We had chosen K’ as 0.15 (see previous page) so we compute S-K’ in cell G3 as F3-$B$3, and then the weight in the portfolio is found by multiplying G3 by $B$6 the weight multiplier computed earlier.

When we follow the steps for all the steps we find that:

• The weights sum to 99.75%, so very close to 100%
• The Theoretical value for the Effective N % is 68.1% and the Realized value is 68.61%

So we have shown that we can generate weights from Sharpe Ratios in very practical way, ensuring that the weights sum close to 100% and the concentration levels of the portfolio are adequate and can be chosen. Of course we assigned the Sharpe ratio as is we were doing a midpoint rule integration but if your Sharpe ratios are truly normally distributed in most cases the above will work, the weights will not sum exactly but the answer will be close. The above ensures that the procedure works well.

Are our weights anywhere close to optimal using our simplified setup?

We are only talking here of ex-ante optimality, ie, if the investor’s view were correct, how close would his weights be to the weights generated by an optimizer seeking to maximise the sharpe ratio of the portfolio under constraints of positive weights and a given maximum concentration.

To answer this we conducted a small scale experiment where we generated the following parameters for our assets, close to what has been seen in the past 20 years in European equities:

-Average volatility and standard deviation of these volatilities

-fixed pairwise correlation close to what has been seen in terms of averages

We varied the selectivity of the investor (where he thinks his investments rank in sharpe ratios ex ante) between the top 10% and the top 40% and generated sharpe ratios for the assets distributed accordingly.

We then used Microsoft Excel’s optimizer (yes that’s not a great tool for that, agreed) to optimize each of the 1000 portfolio which represented a given market scenario, a given investor selectivity, and a given concentration.

How do we appraise the results? I believe the benchmark against which we should measure our weights is the sharpe ratio of an equal weighted portfolio. So the metric I have decided to use is the percentage of the distance between the sharpe ration of the equal weight portfolio and the sharpe ratio of the optimizer derived portfolio. If you are at 0% your weights are no better than an equal weight portfolio, at 100%, you have attained a sharpe ratio equal to what you’d get with an optimizer.

Results:

• The median “performance” ?as defined above of our portfolios with sharpe weights is 91%. This means half of the portfolios using our weights achieved a sharpe ratio that was more than 91% of the way between an equally weighted portfolio and am optimized one.
• 90% of the portfolios are at least 79% of the way between an equal weight portfolio and the optimizer generated one.

This is not very surprising, without constraints, a portfolio in an equicorrelated market would have had weights that were a function of the ratio of expected returns to variance. When we impose concentration constraints, we make those weights closer to sharpe ratios.

?Are these results absolutely certain? No they were established using excel’s optimizer and probably a round of simulation using python and one of the good optimisation libraries would probably be needed. The goal here was to demonstrate that the approach made sense, and probably apart from not being very flexible in terms of available concentration levels, It offers good results for the low level of complexity involved if you want to use simplified inputs to portfolio construction.