Modern Portfolio Theory suggests that every portfolio has its risk and its corresponding rate of return. Markowitz (1952) showed that there is no “best investment” but rather a “best” trade-off between risk and return, called efficient frontier theory.

This theory is based on a critical assumption: the risk and return profile of an asset or portfolio when constructed, is based on historical prices of the assets within the portfolio. As a result, a trader should not fully rely on Markowitz’s theory when making an investment decision; future returns / risk will not necessarily lie on the efficient frontier curve (which is discussed below).

The following formulas need to be considered to understand how the efficient frontier graph is built.

**Return Formulas**

**Risk Formulas**

Now, lets take the following example: Say you have an *X* amount to invest in two assets – stocks and bonds – with equal split. Therefore, you decide to invest in Apple and in the U.S. 10Y Treasury Bill (U.S. government bond). What is the risk and return of your portfolio? What is the optimal split?

Step 1: You can download historical stock prices of Apple, Inc. here and historical Treasury Bill yields here

Step 2: Apply the ln formula. Take ln of the ratio of Price at day i divided by Price at day i-1 applied to the whole sample (in this example the time period is 11 months, 6.1.2016 – 6.12.2016).

Step 3: Calculate the average return and risk of Apple’s stock and TBill’s yield (for average see formula 1 and for risk see formula 4).

Step 4: Calculate excess returns. This is the daily ln return minus the average return applied to the whole sample (formula 2)

Step 5: Construct the covariance Matrix. In this case, since there are 2 assets, it is a 2×2 matrix.

Step 6: Calculate the return and risk of the portfolio using formulas 3 and 5, respectively.

Applying these steps to the example, the portfolio’s average expected return is 8.5% and 1.5%, respectively, based on 50 / 50 split. What would be very interesting to see is how the risk / return relationship changes when using different splits. These are my results using a scenario analysis:

As you can see the inflection point is at a 70/30 split. Take the 60/40 split (1 point below the 70/30 split in the graph). You receive a return of ~8.7% and a risk of 1.4%. However, that would not be a wise investment decision. That is because, in the 90/10 split (2 points above) you can receive a higher return (~9.4%) with the same risk of 1.4%. As a result, in theory you shouldn’t construct this portfolio with a weighting less than 70% of Apple’s stock.

Considering the example above, I will now examine the application of the efficient frontier theory to Biotech. The analysis is done for midcap and largecap firms based on 5-year historical data of stock prices (June 2011 to December 2016). The following 6 portfolio sets analyzed: Mid cap, Mid cap with Treasury Bill, Large cap, Large cap with Treasury Bill and finally, Large cap with Mid cap and Treasury Bill. Two scenarios have been constructed under the latter portfolio: starting with Mid cap weight of 100% and moving down to 0% with a 10% step and by keeping Large Cap weight equal to that of Treasury bill, and the second scenario starting with Large cap weight of 100% and moving down to 0% by keeping Mid cap weight equal to that of Treasury bill.

Mid cap: in this analysis, I have included companies with market capitalisation between USD 1 bn. and USD 10 bn, while large cap are firms with market cap larger than USD 10 bn.

Therefore, the Mid cap sample includes: Alkermes, Jazz Pharmaceuticals, OPKO Health, United Therapeutics, Exelixis, Neurocrine Biosciences, Alnylam Pharmaceuticals, ACADIA Pharmaceuticals, ARIAD Pharmaceuticals, The Medicines Company, Sarepta Therapeutics, Lexicon Pharmaceuticals, Depomed and Emergent Biosolutions.

Large cap: this includes big biotech companies such as Amgen, Gilead, Celgene, Biogen, Regeneron Pharmaceuticals, Alexion Pharmaceuticals and BioMarin Pharmaceutical.

It should be noted that in both the Mid cap and the Large cap samples the weighting is split equally between the different companies.

Applying the same process as described in the initial example, the following results are obtained:

The first observation from these results is that, surprisingly, the Mid cap with Tbill portfolio does not follow the risk return profile according to Markowitz. In fact, the Mid cap graph exhibits a linear risk / return relationship (the actual formula derived from this sample is Return = 13.8*Risk – 0.2).

From a financial aspect, the linear midcap line and the Mid cap with Large cap = TBill graph (purple line) would not be wise investment decisions, as you can get the same risk with a much higher return. Similarly, the Large cap with TBill is a better investment rather than the Mid cap with Tbill rather which is slightly shifted away, exhibiting a higher risk.

Therefore, we are left with the following 2 “optimal” portfolios to choose from:

The shaded red area is a bad investment. Instead, if you decide to invest between the two blue lines the Large cap with TBill portfolio is a clear winner. Above these lines it is an area of personal taste i.e whether you want to invest in a riskier portfolio (Mid cap with Large cap) with higher return a less risky portfolio (Large cap with TBill) with lower return. However, by looking at the actual numbers, at a 100% Large cap portfolio you get the highest return of 18.4% for a risk of 1.4%. If you want to minimize your risk you would need to go down to a return of 11.3% at 1.1% risk (red line, 80% Large cap, 20% TBill). That means sacrificing 7.1% return for reducing the risk by merely 0.3%.

In conclusion, a good investment decision based on the results above, is the Large cap / Mid cap portfolio with the Large cap’s weighting higher than 75%.