Long short (LS) portfolios are not simple to analyze. Their behavior and risk profile can change drastically depending on the composition of the longs and shorts in the portfolio.
Imagine a portfolio comprised of a few technology companies, such as: Google, Facebook, Netflix and Amazon. Further assume that we invest $1000 on each of NFLX, GOOG and AMZN and short $1000 worth of Facebook.
Not surprisingly this portfolio behaves very much like a technology portfolio. The longs are still prevalent and capture most of the risk. Now, if we were to gradually increase the FB short, the portfolio behavior would change. Without resorting to any portfolio software we should expect 3 things to happen:
- Overall portfolio risk should decrease
- The amount of risk in the longs should decrease and in the shorts increase
- The amount of systematic risk should decrease and idiosyncratic risk increase (less and less can be explained by macro factors)
In our 4 stock example the above properties will materialize as FB short increases, but only up to a point. Beyond that point the influence of FB could overpower the risk, and longs would become the “hedge” for the portfolio. Seasoned portfolio managers know this, but how can we attach numbers to this behavior? Looking only at exposures will not be sufficient. One needs to take into account correlations and volatilities as well.
At Everysk we use a measure called Marginal Contribution to Total Risk (MCTR) to express the intricate relationship between assets in a long-short portfolio. Simply put, it is a measure that depends on:
- The amount of money allocated to the position (exposure)
- The individual volatility of the position
- The correlation of one unit of the position to one unit of the portfolio (effectively linking the position to all other positions in the portfolio)
The results for the portfolio above ($3000 long vs $1000 short) are:
|
stock |
Exposure |
MCTR |
SP500 |
Idiosyncratic |
|
(1) |
(2) = (3)+(4) |
(3) |
(4) |
|
|
NFLX |
$1000 |
10.00% |
2.83% |
7.17% |
|
AMZN |
$1000 |
5.89% |
1.88% |
4.01% |
|
GOOG |
$1000 |
3.35% |
2.06% |
1.29% |
|
FB |
-$1000 |
-1.82% |
-2.17% |
0.35% |
|
total |
17.42% |
4.59% |
12.82% |
Some conclusions from this table:
- The total MCTR from longs (MCTR is additive so you can add all longs) is positive an equal to 19.24%
- The total MCTR from shorts is negative and equal to -1.82% (increasing the short will further decrease the portfolio risk)
Those numbers indicate that “longs” retain most of the portfolio risk and “shorts” buffer risk. A simple, intuitive way to understand MCTR’s sign is to remember that it is a product of: exposure x volatility x correlation to portfolio. In the configuration above ($3000 long and $1000 short):
|
exposure |
volatility |
Correlation to portfolio |
Final |
|
|
(1) |
(2) |
(3) |
(4) = (1) * (2) * (3) |
|
|
longs |
PLUS |
PLUS |
PLUS |
PLUS |
|
shorts |
MINUS |
PLUS |
PLUS |
MINUS |
Volatility is always a positive number. Exposure is unambiguous: longs are positive and shorts are negative. The last term reflects the correlation of 1 unit of each security with 1 unit of portfolio. Here both (NFLX+GOOG+AMZN) as well as FB are positively correlated to the portfolio, as the portfolio behaves very much like a tech stock.
Also, if we decompose MCTR into a simplistic one-factor model comprised of: SP + residual (columns 3 and 4 from the first table), we can see that 4.69% is systematic. That represents 26% of the total portfolio risk (26% = 4.59 / 17.42)
Now let’s increase the amount of FB short to $2000. The following table depicts the new state:
|
stock |
Exposure |
MCTR |
SP500 |
Idiosyncratic |
|
(1) |
(2) = (3)+(4) |
(3) |
(4) |
|
|
NFLX |
$1000 |
6.37% |
1.23% |
5.14% |
|
AMZN |
$1000 |
3.23% |
0.80% |
2.43% |
|
GOOG |
$1000 |
1.33% |
0.88% |
0.45% |
|
FB |
-$2000 |
3.08% |
-1.85% |
4.93% |
|
total |
14.01% |
1.05% |
12.95% |
We can see that the overall portfolio risk decreased from 17.42% to 14.01% and the contribution from each stock has changed dramatically. The most drastic change is that Facebook’s MCTR has flipped to positive, i.e. FB will add more risk to the portfolio if we continue increasing the short. We can also notice that most of the risk is now idiosyncratic. The systematic component represents only 7.5% of the total risk (7.5%=1.05/14.01).
The following plots illustrate the changes in the portfolio as we gradually increase the FB short (notice that we also increase the equity in the portfolio to maintain constant leverage):
The x-axis in the graph above depicts the amount of FB short in the portfolio, increasing from $250 to $10000 short (the long exposure is kept constant at $3000). The y-axis depicts the MCTR from the 3 longs (in blue) and FB (in green). The sum of the 2 represent the total portfolio risk (in red).
So, for a FB short of $250 and $3000 invested in the longs, the portfolio risk is 24.8% (the red graph starts at 0.248), whereby 26.3% is coming from the longs and -1.44% from FB.
As we gradually increase FB short, we can see 3 important inflection points:
- The first one comes right away at $1290 when the blue line intercepts the red. At that point all the portfolio risk is coming from the longs. Beyond that point the green line becomes positive (FB stops to work as a hedge).
- The second inflection point happens at $2380 when the blue line intercepts the green. This is the point with minimum amount of total portfolio risk of 13.4% (red at its lowest point in the graph). Contribution from longs and short are the same, i.e. portfolio has achieved risk parity between longs and shorts.
- The third point happens at $4294 when green intercepts red. At that point all the risk is due to the FB short and the longs are not contributing any risk. Beyond that point, the blue line crosses to negative territory (longs become a hedge for the portfolio)
Next we plot similar information, but instead of decomposing risk from the longs and short, we look at systematic versus idiosyncratic:
The axis are the same as our first plot, but now green indicates contribution from idiosyncratic and blue from systematic sources of risk (red is the total portfolio risk just like in the first plot).
It can be seen from the plot that as we increase the FB short, the contribution from systematic sources decreases much faster in the beginning (blue line). At around $2500 the risk has been transferred completely to idiosyncratic. Beyond that point the contribution from systematic starts to influence the portfolio again.
In this technical article we dissected the behavior of a simple long-short portfolio of tech stocks to demonstrate how marginal contribution to total risk (MCTR) can be used to generate optimum risk configurations. We decomposed the sources of risk from longs vs. shorts and systematic vs. idiosyncratic.
Using only simple exposures in the long and short will not capture the subtle correlation/volatility effects described above.