By Francis Marais, 25 July 2016
Needless to say a free lunch would always be welcome, however, to borrow a concept from economics, it has diminishing marginal benefits. The first free lunch is great, not so with the fourth or fifth one. How many different baskets of eggs can you really carry? These examples might be a bit spurious, but it does provide a rather fitting starting point.
At Glacier Research we are fortunate to work with a wide variety of intermediaries, specifically when it comes to investments and portfolio construction. What we have come to notice is that some intermediaries and clients have really taken the concept of diversification to heart. We have come across portfolios that hold between 15 to 20 funds. In some instances these underlying funds might even be FoFs! We believe this is excessive.
In this article we will attempt to look at the benefits and effects of diversification. When does the amount of funds in a portfolio become excessive and what are some of the consequences? We will also look at possible ways to ascertain the appropriate number of funds a portfolio should hold.
Before we answer these questions, it might be a good idea to briefly review the benefits and effects of diversification.
Above is a portfolio possibilities curve for two equity portfolios. This graph indicates that there are efficient and inefficient ways to combine two portfolios. For instance if you start with 100% invested in fund 1, your expected return and risk, measured as standard deviation, is 16% and 10.20%. Adding 10% of fund 2, reduces your expected risk materially to 9.3%, but not at the expense of returns. If each fund were used in isolation, the risk for fund 1 is 10.20% and fund 2, 11.46%. Point 5 represents the portfolio that delivers the highest rate of return (16.025%) per unit of risk (7.79%). Consequently it has the highest Sharpe ratio. This is a classic example of the whole is better than the sum of its parts, and also beautifully demonstrates the benefits of diversification.
Where does this unseen benefit come from? It comes from the interaction of the funds with each other, or in more general terms their correlation. The lower the correlation the greater the benefits of diversification, as each fund reaches the same end goal, but with different strategies. As long as the end goal is the same, the more different their strategies, the better the diversification benefits.
What about adding even more funds? Instead of using two, what is the impact if three, four, five or more are used? This is a relevant question and one we hope to answer in this article. Luckily, we do not have to reinvent the wheel and modern portfolio theory has already provided us with a basis to answer this question through the following formula:
This formula indicates that a portfolio’s variance is equal to the product of the variance and one less the average correlation divided by the number of underlying funds plus the average correlation (CFA Institute, 2013). This formula assumes that the standard deviation of all the funds is the same and that the average correlation between the funds and the portfolio remains constant as additional funds are added. A key interpretation of this formula is that as the number of underlying funds (n) increases the additional benefit relative to reducing the overall volatility of the portfolio decreases. This is known as diminishing marginal rates of returns.
Graphically we can illustrate this as follows:
For instance, let’s take the top light blue line, where the assumption is that the average correlation between individual funds is 0.95. If your portfolio holds only one fund, its variance will equal the underlying fund’s variance. By adding an additional fund you can potentially reduce the portfolio variance by 0.975 indicated by point 2 on this line. Adding a third fund will further reduce the variance by 0.96667 and so forth. Using less correlated funds with an average correlation of 0.8 (the lower yellow line) will reduce the overall variance by 0.9 (as opposed to only 0.975 in the first example where the average correlation was 0.95). Adding a third fund further reduces the variance by 0.86667 (0.96667 previously) etc.
What is clear from this graph is that:
Additionally the lower the correlation between two funds, the more significant the diversification benefit. However this decreases at quite a significant rate and by the time the fifth fund is added most of the benefits have been achieved. By the time the tenth fund is added there are very few benefits left in reducing the portfolio’s overall variance.
Another interesting point to consider is that as the number of funds (n) gets infinitely large, the term (1-ρ)/n gets infinitely small and the reduction in portfolio variance will converge to the correlation between them. For a group of funds with a 0.95 correlation between them the maximum reduction in portfolio variance will be 0.95.
So to recap: The lower the correlations, the more funds you are able to combine with better diversification benefits. However, the benefits of diversification decrease per additional fund added to the portfolio.
Let’s move from a theoretical to an empirical example and look at some actual funds and what happens when we combine them in portfolios. In these examples we will start with the fund that has the lowest average correlation and add the fund with the next lowest average correlation. We continue doing this until all the funds form part of the portfolio.
For the first example, we will use a basket of 20 equity funds with at least a five year track record. These funds had an average correlation of 0.86 relative to each other. Hence there is potential to decrease portfolio variance by adding additional funds to a portfolio consisting of only one equity fund.
Starting with a single equity fund and adding an additional equity fund significantly reduces overall volatility (as measured by standard deviation). This continues as additional funds are added until there are five funds in the portfolio. (See circled area in the graph below.) After this point, adding additional funds actually increases the portfolio’s volatility. Why is this? As one continues to add funds, the portfolio’s correlation to the other funds starts increasing. Therefore as funds are added with higher volatility and higher correlations, the overall portfolio volatility starts to increase until it reaches a point above 0.95. Soon after this point is reached, overall portfolio volatility starts trending upwards as funds with higher volatilities are added. An important observation is that a portfolio consisting of all 20 funds realised a portfolio volatility of 9.34%, the highest of all the other portfolios containing fewer funds, but still substantially lower than the average volatility of the individual funds of 10.13%.
So the next important question is what happens to the overall portfolio’s return? It initially increases, but then decreases and converges around 15.50%, as more funds are added. (See graph below.)
With volatility trending upwards and the return converging around 15.50%, risk-adjusted returns therefore deteriorate as expected. See how the Sharpe ratio in the graph below declines as more funds are added to the portfolio.
What is also not entirely unexpected is that the overall correlation with the broader JSE also increases. So in effect overall systematic risk, or broader market risk, has started to increase. While you might have reduced idiosyncratic manager risk, you have started to increase overall systematic risk, leading to higher total risk as measured by standard deviation. (See graph below.)
Instead of only focussing on similar types of funds, what would happen if we combine different types of funds across different ASISA categories? A combination of funds that includes money market, pure bond funds, pure property, pure equity and some multi-asset funds would have lower average correlations and subsequently the potential for better diversification benefits.
From the graph below one can see that combining different types of funds does indeed offer good diversification benefits. However after approximately 10 funds, the overall portfolio volatility once again starts to increase.
In addition, portfolio return converges and consequently risk-adjusted performance as measured by the Sharpe ratio again, starts decreasing. (See graph below.)
From the data and findings presented above one can conclude that adding more than 10 funds to any portfolio results in very little diversification benefits and in some instances might even result in higher overall portfolio volatility. Are there certain tools one can employ to assist in deciphering when a fund should be added and when enough, is truly enough? The answer is yes and it lies in modern portfolio theory. The addition of another asset to a portfolio is optimal if the Sharpe ratio of the new investment is greater than the Sharpe ratio of the current portfolio, multiplied by the correlation of the new investment with the portfolio [CFA Institute (2013), Elton, Gruber and Rentzler (1987)]. This is a relatively easy method and can be implemented with ease by most investors and intermediaries. Let’s put it to the test by using the same universe of funds previously used.
We will start by focussing on the portfolio of equity funds. If we have one equity fund and only add a new one if the above rule holds, then we will arrive at a portfolio with five equally weighted funds. After this the rule does not hold for any fund anymore. See the graph below which shows the total return of this portfolio is 16.87%, with a standard deviation of 8.38% and a Sharpe ratio of 1.32.
Adding funds together in this manner leads to a significantly higher Sharpe ratio than we were previously able to realise when we just focussed on low average correlations. According to this rule and the universe of funds, one cannot add a sixth fund as it would lead to lower risk-adjusted returns. The graph below indicates how the portfolio benefits from adding additional funds until the portfolio consists of five funds and also what happens when a sixth additional fund is added to the portfolio.
What about funds across different categories? Due to the lower average correlations we are able to utilise a larger number of funds, but diversification benefits seem to stop once the portfolio has approximately nine funds. In this instance the Sharpe ratio we are able to realise using the decision rule is 2.33, with an expected return of 18.92% and a portfolio standard deviation of 5.64%. (See graph below.) The Sharpe ratio in this instance is slightly lower when compared to our previous combinations focussing purely on average correlations. This emphasises the fact that the order in which you choose to add different types of funds becomes very important including your starting point and the level of risk you would like to add to your portfolio, if the underlying funds differ significantly from each other.
In all the previous examples the funds were equally weighted. Secondly, using Sharpe ratios and correlations only helped determine if a fund should be added to the portfolio or not. It did not indicate how much the allocation to each fund should be. Another alternative is to add the different funds into a mean-variance optimiser. This is essentially a computer programme with the capability to run iterations and look for the best possible combinations of the different funds according to a specified goal and constraints. If our goal is the highest possible risk-adjusted returns with the constraints that the individual fund weights cannot be less than 0, all the weights need to sum to 1 and the weight to any single fund cannot exceed 20%, we get some interesting results.
In the above graphs we summarise the results from adding portfolios together using different decision rules, i.e. average correlations, a decision rule using Sharpe ratios in conjunction with correlations and the output of the optimiser.
What is interesting from these two exercises is that in each instance the optimiser chose a number of funds far less than 10. A further insight already mentioned is that due to the higher correlations, when exclusively using equity funds that are similar, the lower the number of funds included in the portfolio. Additionally in no instance were we able to justify the inclusion of more than 10 funds in a portfolio.
We have shown what diversification is, more importantly the effect of lowering overall portfolio volatility without necessarily compromising returns. We have also shown the effect of adding additional funds and the diminishing effect of this as well as the importance of correlations when it comes to portfolio volatility.
We would like to end off by stressing the absolute importance of diversification. This article should not be construed as an indictment on diversification. However, in a relatively small, open market such as ours an excessive number of underlying funds in a portfolio might not be optimal and can actually increase overall portfolio risk. While these are interesting insights, they should be used to enhance your overall portfolio construction process, but not undermine the diversification benefits. It is our opinion that the risks of over diversification are far less than the risk of under diversification. This is the final point and thought we would like to emphasise and leave with you.
CFA Institute (2013). CFA Program Curriculum, Level II, Volume 6, Derivatives and Portfolio Management, 378.
CFA Institute (2013). CFA Program Curriculum, Level II, Volume 6, Derivatives and Portfolio Management, 391.
Elton, E.J. and Gruber, M.J. and Rentzler, J.C. (1987). Professionally Managed, Publically Traded Commodity Funds, The Journal of Business, 60(2), pp. 175-199.