**Introduction**

RisCura has often indicated the importance of diversification in portfolio construction. This view, whilst generally accepted for institutional investing by fiduciaries looking to exercise their duty of due skill, care and diligence, is often challenged with (at least) two counter assertions:

- Active managers believe they thrive in a world of volatility, through focussed bet sizes, capturing this volatility in portfolio alpha or outperformance.
- Many investors think one can over-diversify a portfolio to the extent there is no possibility for alpha.

The first point above was famously backed by Warren Buffet, who indicated that really good professionals sought volatility, and for everyone else he advised diversification. Since for institutional investing diversification is promoted as one of the principle ways of reducing portfolio risk, this document focusses principally on the second point and using a discussion of the evolution of portfolio theory progresses this discussion.

**Modern Portfolio Theory**

The traditional text for portfolio construction learnt in all first year finance courses arises out of Modern Portfolio Theory (MPT). The following is Investopedia’s description:

“Modern portfolio theory is a theory on how risk-averse investors can construct portfolios to maximize expected return based on a given level of market risk. Harry Markowitz pioneered this theory in his paper “Portfolio Selection,” which was published in the Journal of Finance in 1952. He was later awarded a Nobel Prize for his work on modern portfolio theory […] Modern portfolio theory argues that an investment’s risk and return characteristics should not be viewed alone, but should be evaluated by how the investment affects the overall portfolio’s risk and return within a discrete period. MPT shows that an investor can construct a portfolio of multiple assets that will maximize returns for a given level of risk. Likewise, given a desired level of expected return, an investor can construct a portfolio with the lowest possible risk. Based on statistical measures such as variance and correlation, an individual investment’s performance is less important than how it impacts the entire portfolio.

MPT assumes that investors are risk-averse, meaning they prefer a less risky portfolio to a riskier one for a given level of return. As a practical matter, risk aversion implies that most people should invest in multiple asset classes.”

Note that most institutional investors by virtue of their fiduciary duty, are volatility averse and so MPT is commonly understood as a basis for institutional investment practices.

**Portfolio Implications and Assumptions of MPT**

Investopedia goes on to add: “The expected return of the portfolio is calculated as a weighted sum of the individual assets’ returns. If a portfolio contained four equally weighted assets with expected returns of 4, 6, 10, and 14%, the portfolio’s expected return would be:

(4% x 25%) + (6% x 25%) + (10% x 25%) + (14% x 25%) = 8.5%”

The above is a relatively simple mathematical formula and derives an intuitive result.

When it comes to calculating the risk, the mathematics becomes a little more complicated. We pick up with Investopedia as follows: “The portfolio’s risk is a complicated function of the variances of each asset and the correlations of each pair of assets. To calculate the risk of a four-asset portfolio, an investor needs each of the four assets’ variances and six correlation values, since there are six possible two-asset combinations with four assets. Because of the asset correlations, the total portfolio risk, or standard deviation, is lower than what would be calculated by a weighted sum.”

The formula for portfolio variance in a two-asset portfolio is:

When we look at the last term of this portfolio variance equation, we see that the higher the correlation between the assets, the higher the variance. It stands to reason that, if one can find assets that are relatively uncorrelated, then all else being equal, one can construct a portfolio with lower volatility. In the case where the assets are perfectly correlated, the portfolio variance would be the square of the weighted sum of the share risks, plus the final term in the sum above, which in that case would be at its maximum. In other words, when assets are perfectly correlated, the portfolio variance is at its highest.

What we could conclude from the above examination of modern portfolio theory, is that to the extent that diversification introduces uncorrelated assets, it impacts risk by lowering it. On the other hand, portfolio performance or return is directly influenced by how the introduced asset performs.

Of note with MPT is that it concentrates on performance and portfolio holdings in one period only i.e. a period, where the return and portfolio holdings are only dependent on the portfolio inputs (holdings and returns) for that period only. So, as we saw above the return in the period is simply the weighted sum of the returns of the underlying holdings. This is represented diagrammatically as follows:

**Stochastic Portfolio Theory**

Stochastic Portfolio Theory (SPT) is a mathematical theory for analysing stock market structure and portfolio behaviour discussed by E. Robert Fernholz in 2002. SPT uses continuous-time random processes (in particular, continuous semi-martingales) to represent the prices of individual securities. So, in essence, this theory allows one to look over time, or multiple periods, at the impact of changing holdings, each with their own probability distributions, in a portfolio.

Using logarithmic returns, one gets the following long-term portfolio growth rate, over multiple periods:

What this implies is that with diversification, a portfolio return outperforms the underlying components of return by the “Excess Growth Rate” over the long-term.

**Portfolio Implications of SPT**

This newer approach to portfolio construction had a key implication on diversification. To provide more colour:

The excess growth rate is as calculated above, where the average stock volatility is the linear weighted sum of the share volatilities, and the portfolio volatility is as described above in MPT. As we saw above, portfolio volatility is usually lower than the sum of volatilities for any assets combined which do not have perfect correlation.

#### Implications

This excess growth rate term in the formula has the following theoretical implications for portfolio construction:

- Portfolio return in the long term is higher than the sum of the individual returns over that period, as assets are generally not perfectly correlated.
- The excess return can be quantified in return ‘units’ or percentage performance. i.e. this is not just conceptual debating of obscure concepts.
- Diversification is not just about risk minimisation as with MPT, but also return enhancement. In other words, it pays ‘return’ to diversify.
- The higher the variety of investment propositions in the portfolio, within each asset class and in totality, the higher the long-term performance. This presupposes that the additions have the same quality of return over the long term. The additions should still be good, in order to keep the expectation of performance similar.
- Even if a small allocation is made to the portfolio of an asset or mandate with higher return expectations, and even if it has higher risk, as long as it is uncorrelated to the balance of the portfolio, it will add return value to the portfolio over and above its weighted contribution.
- A more diversified portfolio will always result in a positive contribution to performance.
- The more volatile markets are, and if components across and within markets have high cross-sectional volatility, the more the value added by diversification.

**Conceptually, why is this happening:**

- Volatility in any single asset price acts to lower total return over time. Put another way, geometric compounding does not like volatility. Meaning, the goal of the diversification should be to stabilise performance, ideally at higher levels over time.
- Each asset in a portfolio has a distribution of potential returns and its risk/volatility. When we look in any one period, each asset will have a return that falls somewhere in that distribution. With only a few assets in the portfolio the risk exists that all assets in a particular period, despite their correlations, may be on the downside, or the upside of those distributions. As you add more and more assets, and more and more time periods, the aggregate return of the portfolio will tend towards the weighted sum of the averages of all the underlying assets, and this will lower the volatility of the returns that need to be compounded, and hence add value to the portfolio given 1 above.
- Rebalancing plays a component with these diversified portfolios and also adds value.

#### Practical considerations

In practice there are some considerations we would also point out:

- As with any theory, there is slippage between the theory and the practice. SPT has thought through some of these and processes with discontinuities, such as jumps, have also been incorporated into the theory.
- Diversification may add additional cost to the running of a portfolio as mandates and assets added to a portfolio may make the portfolio management more expensive.
- A caveat, for liability-driven investment institutions such as pension funds, is that all the formulae above should be contemplated relative to liabilities. In this way, interest rate and inflation risk are controlled in a premeditated manner, and diversification into performing uncorrelated assets in the portfolio adds value.

**Conclusion**

The document examines the benefits of diversification through the lens of portfolio theory. It provides a brief overview of developments in the portfolio optimisation space. Newer theories, SPT in particular, put together based on multiple periods of portfolio optimisation, with less restrictive descriptions of the components of the portfolios, point to benefits of diversification that go beyond assisting with risk management.

Under this evolved view of portfolio optimisation, diversification and rebalancing over time enhance portfolio performance, over and above the commonly understood risk reduction they bring. Other interesting implications are as follows:

- In theory, more diversification is better.
- We need to consider diversification and portfolio construction over multiple time periods.
- The implications over time can be quantified in units of additional performance which can be measured.
- The additional performance is enhanced the more volatile the markets and assets are, and the higher the cross-sectional volatility of the assets are.

Finally, in a liability-driven investment environment, whilst interest rate and inflation risk should be controlled in a premeditated manner through strategic asset allocation, diversification of holdings during the portfolio construction phase adds value to performance.