# Diversification’s Failure and the Need for Non-linearity – Full Article

#### Dr Michael P. Streatfield, CFA

Quantitative strategist, STANLIB Absolute Returns

#### Muntu Mdwara

Quantitative analyst, STANLIB Absolute Returns

##### Summary

Diversification is lauded as the one free lunch of investing, and this certainly helps to manage risk. However, relying solely on “average” correlations between assets can leave the investor served with measly gruel when they most need nourishment.

We look at the challenges in markets this year for South African investors, and question the reliance on equity – bond correlations that drives most traditional balanced fund asset allocation approaches. This was little comfort in March 2020 as both local and global equity and bond correlations spiked, and drawdowns from the South African Retirement Funds Domestic and Global sector rose to double digits.

As the STANLIB Multi-Strategy team, we believe all assets are risk assets. Buying bonds for ‘safety’ does not take into account asset behaviour in extremes, nor does it draw on other lenses, like volatility.

We look at the risks in the tails and find that, at times of big drawdowns, equities and bonds do not always behave as expected. Conditional correlation approaches confirm the non-linearity of the correlation between assets in down markets in extreme conditions.

Diversification is not hedging. By this, we mean hoping to manage risk by holding a variety of assets is not the same as hedging (which explicitly targets a risk and protects against it at some cost – financial or foregone return). We look at the historic costs of simple equity hedging strategies and find the penalty on returns is large, and tail protection is not guaranteed.

We explore ways that more sophisticated multi-asset managers can improve outcomes by understanding volatility regimes and assessing correlations in a more tail-aware way, and we demonstrate the benefits of an active approach to hedging.

**The implications for retail investors and fiduciaries are:**

The traditional balanced fund approach of simply holding bonds is not a guarantee of safety. In extreme market conditions, conditional correlations between equities and fixed income can rise, as we saw in March 2020. Bond funds are not the only answer to manage risk.

Hedging tail risk is one way of managing tail risk, but a passive systematic ‘always putting your foot on the brake pedal’ approach is very expensive, both in return upside foregone, as well as the actual risk in implementation. High option premium costs mean permanent hedging or structured products can suffer large return drags.

There is a growing need for active investment management approaches to choose non-traditional ways to tactically deal with non-linearity – choosing the appropriate conditions and execution for hedging.

More than one lens is needed to view the market. Volatility conditions can provide insight into when hedging can be more cost-effective.

In very benign markets, equity hedging eats up returns and downside risks are low. In highly volatile markets, when fear is already high, it is not the time to hedge. Protection costs are excessive. With the inverse relationship between VIX changes and equity returns, downside risks are in the rear-view mirror and prospective forward returns look good. It is really in the “Goldilocks” times that dynamically hedging can take the sting out of the drawdown tail.

Multi-asset approaches need to evolve, both in how managers address diversification in extremes and how they purposefully hedge tail risk. Multi-strategy is such an evolution.

##### Introduction to Bond Equity asset correlation breakdown

The core tenet of a traditional balanced fund (60/40) is a combination of equity and fixed income (bonds) to manage risk across business cycles. The investor hopes that in times when equity does badly, bonds will step up and provide defensive support.

In this year’s equity collapse, Figure 1: Retirement Fund Returns Year-to-date shows this was certainly not the case as the COVID-19 pandemic gained momentum. In the first quarter of 2020, returns for South African retirement investors in both the domestic-only (SA Large Manager Watch) and global (balanced with international) portfolios collapsed and had still not recovered four months later. Drawdowns of double digits were experienced, even for the global “more-diversified” portfolios. (Both median returns and average returns followed similar patterns.)

The dynamics behind this weak performance can be seen in Figure 2: Equity Bond Daily Correlation. In this figure, the top panel is the SA Equity market, proxied by the Capped SWIX. The correlation with the All-Bond index is in the middle panel, with the correlations to US Equity in the bottom panel. In the middle panel, the teal ellipse highlights the time of equity distress in March 2020, when correlations to bonds surged above the past five-year average of 0.29 to 0.85 at the peak. In other words, as equity prices fell in the quickest bear market ever, bond prices did so almost as rapidly. The bond assets did not move in the opposite direction, and diversification failed.

*This daily data example uses 60-day correlations, to approximate a quarter, over 5 years of data. Source: Bloomberg, GCOR.*

Even over the monthly time horizon (Figure 3), two-year bond-equity correlations in both the domestic and international markets oscillate a great deal. This ‘notion’ of fixed income offering protection in distressed equity markets is fading away, given the number of times the correlation has flipped over the years.

*Figure 3 shows 24 months rolling correlation using 15 years of monthly return data to calculate rolling correlations between Bonds and Equities.*

Strategic asset allocations exercises that focus on very long (five- or ten-year) correlation relationships build a foundation on these false hopes. Shorter-term estimates can be more responsive and may serve as warning signs for regime shifts.

We have discussed single bond-equity pairs, but we can also evaluate correlation between these different asset building blocks. Figure 4: Cross Asset Correlation illustrates how various asset classes historically correlate with each other within a hypothetical multi-asset portfolio. The average level of correlations across asset classes has more than doubled from around 20% before the COVID-19 pandemic to roughly 60% over the last six months.

*Figure 4 shows illustrative cross-asset correlations in local currency between four broad asset class indices: Global Equity and Fixed Income, SA Equities and SA Bonds over rolling 24 months using 15 years of monthly data.*

These recent market events highlight the costly impact that improbable, or highly unlikely, events can have on investors’ portfolio diversification ability. One or two selloffs do not necessarily constitute a trend, but do they signal investors should consider the potential scenario of the disappearance of classic bond-equity hedge behaviour? We will show the non-linearity of correlations in these extremes. This puts long-term investors under significant pressure to find alternative sources of diversification.

##### Exploring the 60/40 Portfolio Downside

Investors in practice are more concerned about tangible changes in their wealth, not mathematical relationships they cannot see, like correlations. We also know from behavioural finance that investors are more sensitive to losses. So, we explore what losses a typical 60/40 balanced with international portfolio would suffer, using historical asset experience.

In this simple exercise, we create a portfolio using total returns from four broad asset class indices. We use the JSE total return history, which is why the period is 18 years. This is a costless, hypothetical portfolio in which we rebalance to static weights on a monthly basis. The portfolio uses the current 30% offshore allowance and is split 60/40 between equities and fixed income. Active balanced managers could argue that they would add additional value through asset allocation and stock selection. But in reality, there would also be investment and trading costs to consider. Exchange control levels have changed during this lookback period. The goal of this exercise is to get an empirical idea of what an investor’s regulatory portfolio today might experience, drawing on asset performance from SA’s history.

This non-normality of domestic investor returns can be visually seen in a Q-Q plot (or quartile-quartile plot) where a distribution’s quartiles are being compared with each other, in this case an empirical distribution of returns against what would be expected from a normal distribution. The red circle in Figure 5: Q-Q Plot of Hypothetical 60-40 Portfolio shows the large deviation in the left-hand tail from what is expected (which would be the purple straight line if distributions were identical).

*Figure 5 shows the quantile to quantile plot of plotting the empirical monthly return quantiles against a normal distribution quantiles. If they were from the same distribution the points would be on the purple line. The actual distribution of the returns can be found in the Appendix.*

We find even in a classic diverse, balanced with international portfolio the downside tail is fatter than a normal distribution with more negative returns. This points to more potential losses for investors even with the equity-bond diversification.

We outline the magnitude of such losses expected from history, showing the top five drawdowns of this hypothetical portfolio, and what has happened to the building blocks from the start of the drawdown period in Table 1. The hypothetical third drawdown -14% compares well with actual returns experienced in 2020 from Figure 1: Retirement Fund Returns Year-to-date of around -13%.

Looking at these five portfolio drawdowns, you can see the individual asset class performances from the start to the trough of the drawdown in Table 1. The negative performance of South African bonds in the latest drawdown (the third one) clearly stands out. In the Offshore Bond space, the bonds only provided positive performance in three of the five drawdowns.

This reinforces our point that a passive holding in fixed income does not guarantee protection in extremes. This sets the scene for the need for investors, despite geographical and asset class diversification, to contemplate downside risk management in these larger tails to manage these large drawdowns. We now look at conditional correlation and the role of such hedging.

##### What correlation counts?

**Correlation in the Tails**

Investors are worried about adverse outcomes, and loss aversion makes the pain of large losses sting even more. Focusing attention on these tails, the extremes of the return distribution, means that we can get a better understanding of how assets are behaving (and in particular co-behaving) at times of distress.

How effectively an asset performs as a hedge can change, depending where you are in the return distribution. Conditional correlation means assessing correlations assuming a certain market condition is in place – like weaker equity markets. But tails, by their nature, are at the extremes, so there is less data to make accurate assessments. Fortunately, statistical techniques can help to boost this to get an idea of conditional correlation.

Flexible probability estimation allows one to get return estimates focusing on specific states and/or time conditions. This estimation technique makes broad use of the whole data distribution, but re-weights the observations based on the state of the data and how recent it is. For more information on flexible estimates see Flint, Seymour, and Chikurunhe (2019) or Page and Panariello (2018). We use this to estimate conditional correlations.

In Figure 6, we can see the conditional correlation of how assets correlate to weak US equity markets (taken as returns in the lower quartile), and the variation of this change in the bad times (the left tail) and the good times (the right tail). For example, when you consider the full sample of history over more than a decade, South African bonds (ALBI) in green appear to be offering some diversification, with 30% correlation to US equities. However, in times of crisis, these correlations rise on both good and bad extremes, lowering diversification. What is most concerning is that in equity bear markets, the bond markets correlation can rise to a whopping 87% at the 1% percentile. Diversification is not always available when investors need it most.

*Source: STANLIB Absolute Returns, Peresec. Conditional correlations calculated assuming the S&P 500 (SPX) is below 25% percentile threshold. These are flexibly estimated using exponentially weighting with a half-life of around 5 years. The Left Tail and Right Tail are estimated from the 1% (and 99%) of the conditional distribution. Data period Dec 2009 to Jun 2020.*

This is why the philosophy of the STANLIB Absolute Returns team is that all assets are risk assets. A multi-asset investor cannot add a defensive building block, and assume this has ticked the ‘safety’ box. A deeper appreciation of asset dynamics is required.

People tend to think in linear relationships, but at the ‘edges’ markets can behave in non-linear ways. Figure 7 demonstrates this asymmetry. Here the conditional correlation at all percentiles is charted, and shows more than just the 1st and 99th percentiles shown above in Figure 6.

This demonstrates that the conditional relationships curve at the tails and can change direction from what is expected. Both global and domestic bond correlations are actually rising in the tail, not falling, as traditional investors ‘hope’ from their bond component. The non-normality and non-symmetry of assets at extremes cannot be wished away.

##### Costly hedging?

One approach to manage tail risk is to hedge the risk by paying for protection. In this section, we will explain that equity hedging is costly. Estimates of even highly-liquid US equity three-month option premium costs can average 1.5%, but the cost range has large positive skew and can go as high as 3.5% to 4.5%. We believe that a passive approach or mechanical mindset of rolling can be expensive, and is not the optimal solution. An active approach is needed to best minimize hedging costs.

Today an increasing number of investors, prompted by memories of the Global Financial Crisis (GFC), have flocked to options as a way to hedge the tail of their portfolios. This has created a high demand for puts, resulting in a steady increase in put prices. For example, in October 2020, a 5% out-of-the-money put option on the S&P 500 index expiring three months from today would cost over 3.2% (indicative pricing). This premium implies a break-even rate of over 8.2%, in other words the US equity market would have to fall by more than -8.2% in three months before the hedge becomes profitable for an investor. This is very expensive by historical standards and could be far beyond what many rational investors would be ready to pay.

The position was even more extended six months ago. To illustrate the variation in option prices, we source option prices from Bloomberg and express the ranges in Figure 8 for the US and South African equity markets. These processes are for illustration purposes only, as we did not account for any bid-ask spread or other trading costs associated with buying put options on these respective indices.

This cost range denotes hedging is typically not cheap and should not be treated as a free lunch to eradicate market downside risk. Investors need to be cognisant of the drag effect these costs could have on their fund performance.

*Source: Bloomberg. Figure shows range of option premium prices quotes over ten years for the S&P 500 and the JSE ALSI 40 Index. The option selected as a standard 5% out of the money put with a three-month expiry.*

A very popular hedging strategy among investors, due to its simplistic nature, is a protective put strategy. We focus on this protective put strategy to illustrate the impact of hedging. (For those less versed with options, there is more detail on the payoff profile in the Appendix.)

A protective put option is created by buying the stock or an equity index, and buying the associated put options. The protective put is also known as a synthetic long call as its risk/reward profile is the same as that of a long call. The idea of a protective put is to hedge against adverse market movements, not to realise profits from strategy, which makes this strategy equivalent to an insurance policy, rather than a speculative investment. Protection kicks in when losses are below a certain strike level. The total return of the strategy is reduced by the cost of the put options and the price path of the underlying assets (in our case the S&P 500).

What an investor receives is shown in Figure 9. The protection has removed the downside tail risk but this has a cost to the insured portfolio. The probability of larger positive returns is now much smaller due to the cost of this protection. If no adverse returns are experienced, then the investor will have lower returns than the uninsured investor.

This seems good in theory but we will show that a naïve hedging approach (or systematic approach) is not a clear-cut solution for managing tail risk in a portfolio.

The flaw of buying put options on a mechanical basis (rolling puts on a monthly or quarterly basis) is that it does not necessarily hedge the downside risk as expected. We simulate such a strategy to see how it performs in practice. We see this mismatch of expectations in the left-hand tail depicted in Figure 10, where the unhedged strategy outperforms the hedged strategy (see red ellipse).

*Figure above looks at the empirical performance of a hedged and unhedged equity strategy using actual option pricing data on the S&P 500 Index rolled on a monthly basis. Date from June 2002 to Sept 2020. Source: Bloomberg. The hedged performance in the left tail is worse in practice than what one expects. The rolling costs of protection has penalised returns.*

The reason for this mismatch is, firstly, because equity (S&P 500) drawdowns do not necessarily coincide with an option’s expiration cycle. The defensive nature of the put can then become inadequate. Secondly, buying an equity put option reduces the portfolio’s equity exposure, which is like disinvesting your equity component (like holding cash). Both risk reduction approaches (holding cash or buying puts) reduces your equity risk and reduces your expected return. A key fundamental difference is that the put option introduces a time-varying equity exposure that helps to hedge the tail risk. The path dependency arises from the fact that the put option must be rolled over as it matures into a new option at prices that will depend heavily on the underlying asset prices at expiry and the prevailing implied volatility.

In practice, if markets have fallen, then implied volatilities will be elevated, making rolling the protection more expensive at that point.

Hedged strategies do, however, help to cushion large falls, as you can see in Figure 11: Drawdown Comparison Hedged and Unhedged Strategy. Even a 5% out-of-the-money protective put made a huge change to the maximum drawdown in the Global Financial Crisis. It is during these extreme tail events that optionality plays its role. So, there is a case for tactical hedging, although continuous hedging is not practical.

Another important issue for hedging structures is the issue of trade sizing (i.e. how much of the book to hedge). Table 2 depicts the annualised returns and maximum drawdown data for different put structures on a quarterly allocation since 2009. It is important to have sufficient hedge budget (risk budget), otherwise a tail hedge may not deliver the required results. Seemingly high hedge allocation can translate to a drastic drag on performance (looking at CAGR vs. S&P 500) plus the protection on the downside is relatively minute, if a systematic approach is applied.

*Table shows the performance shortfall (CAGR Shortfall) annualised drag from the S&P 500 return. For example, a 25% allocation to a 5% OTM Put will cost -1.6% p.a. giving only 9.3% return.*

Given the challenge of grinding option costs and the practicalities of applying protection, investors need an active approach to risk management. Investors need to step back and choose when to add protection. STANLIB Absolute Returns uses a lens approach, in which we look at markets through different lenses. One of these lenses is volatility. We will show that the different volatility regimes, as assessed from the VIX, can provide vital information on the forward risk (as measured by future drawdowns) depending on the regime. Although the devil is always in the implementation details, this regime can signal when it might be more economical to hedge.

##### The state we are in

Volatility, or the rate of change in prices, provides valuable information on the state of markets. This is why it is one of the STANLIB Absolute Returns lenses.

The most liquid market for trading volatility is the S&P 500. In Figure 12: Inverse Relationship of S&P 500 and VIX, we can see the inverse relationship between the VIX Index and the S&P 500. When we see large positive changes in the VIX Index, we can expect the S&P 500 has negative monthly returns (green ellipse below).

*Figure shows the S&P 500 monthly returns against the VIX. These are concurrent monthly changes over last 18 years on concurrent basis.*

We explore this relationship by looking at different regimes of the VIX and what relationships it holds for future returns. In particular, we will show that the potential drawdowns for investors can greatly depend on the state of market volatility at the outset. This holds important information for the timing of hedging.

*Table shows forward returns on S&P 500 over period January 1990 Through Aug 2020. Daily data used.*

We compared the level of the VIX with subsequent 12-month relative returns on the S&P 500 Index over rolling 12-month periods from January 1990 to August 2020 (Table 3: Historical Scenario Analysis of VIX Regimes). Returns were sorted into three buckets based on initial VIX levels. A “low” VIX was defined as being in the bottom quartile of all readings over the period, “medium” VIX levels fell in the middle two quartiles, and top-quartile readings were considered “high” VIX levels.

This analysis shows that the forward returns expected from the VIX regime vary a great deal per regime. There is clear asymmetry between risk and reward, depending on the VIX category. The average forward return in a High Regime is the lowest at only 6.1%, but the downside risk is very high with a drawdown of -24% at the 10th Percentile returns. However in benign times, Low VIX, the forward returns are higher and downside risk is low at -1.17%. It clearly does not make sense to hedge in this Low state. The chance of adverse returns is lower, so it is not worth giving up upside with a protection strategy.

The VIX index evolves stochastically through time and it exhibits relatively persistent changes of level due to news and/or financial crises. We thought about this stochastic process and looked at modelling VIX to identify the latent hidden states between regimes. This modelling took the process further than just splitting into past percentiles, but it took a data-driven approach to identify when there are distinct risk and return clusters in the data.

To take account of this behaviour, we present a regime-switching model to characterize the evolution of the VIX index.

Applying Machine Learning techniques, we were able to find the hidden volatility regimes, namely: State 2: Normal Regime, State 0: Low Regime & State 1: High Regime. Volatility tends to cluster in a certain way until it switches to a new state. These states are identified by a Hidden Markov Model which determines these distinct clusters. Figure 13 shows the VIX regimes that characterise the path of the VIX Index.

*Figure above shows the VIX states given the risk and return clusters identified by the Hidden Markov Model.*

##### Actively Hedging with the Volatility Lens

We returned to the hedging exercise, using protected put data from June 2002 to September 2020, to demonstrate the benefit of using the VIX State to guide hedging. We have demonstrated earlier that hedging costs are high, investors are concerned about non-linearity and extreme tail risk, and there is a cost-benefit to hedging given this return drag.

We pull this together by using the VIX State to time when to hedge the portfolio. Table 4: Hedging Analysis by VIX states shows these results.

Note in the first line the costs of hedging (how much the forward hedged portfolio return is less than unhedged return) varies a lot by regime. In the High state hedging the portfolio takes 7.8% from the average return. Only in the Normal state does the average forward return benefit from hedging.

Figure 14 illustrates the drawdowns per regime. Here in the Low State, where we would not recommend hedging, the protection costs leave the investor with a worse return than the unhedged portfolio and a lower return. Recall that in this state downside risk is lower, so the drag of protection curtails the return distribution. The drawdowns are not that bad at the 5^{th} percentile, which makes the need for hedging less imperative.

*Figure shows data from Table 4. Range of returns are 90 ^{th} to 5^{th} percentile.*

In the High State, the market is volatile and here people are very concerned about protection. But this is time to be greedy not fearful. Putting on protection is expensive and would cost you an average 7% in alpha (for a mere 9% worst-case drawdown). Hedging in the High State is only rewarding to a very risk-averse investor concerned about any losses. Here an active approach of putting on the protection in this state positively curtails the worst return. It is now positive 5.65% versus the unhedged -9.84%. There is a very large cost to this, as can be seen in the best return possible (see orange brackets).

Paradoxically, the biggest returns from hedging being used to lower the worst return were in the normal state, with a modest cost to the average forward return (see orange ellipse).

Bringing in another signal, in this case the volatility regime, has a large influence on the benefits of hedging. One needs to assess the limits to the upside contrasted with the actual protection in the tail. This is why we do not advocate a systematic hedging approach, as the costs are non-linear, nor are the risks and the upside that portfolios face.

In practice, active managers add further depth and sophistication. They will choose when to put on the protection, agree on a budget for the inevitable drag of protection on the return, make informed calls on which contracts to buy (given the volatility term structure), and what derivative strategies to put on (e.g. put spreads). This will also be done in the context of the other portfolio assets and market conditions.

We used options here as an example of dealing with non-linearity, but it is not only tied to derivatives. Other approaches to manage downside risk are non-traditional assets such as risk premia, private equity, hedge funds, FX or volatility strategies. These all have pros and cons but can be proactive ways of addressing the bond equity correlation in extremes.

In closing, we have shown an active approach to choosing protection, and being guided by the volatility lens, can provide a more tactical approach to providing peace of mind to investors than relying on simple correlation.

#### Appendix

##### References

Flint, Emlyn James, Anthony Seymour, and Florence Chikurunhe, 2019, Estimation with Flexible Probabilities: Identifying Rand Hedges, Finding Diversifiers, Enhancing Style Analysis *Peresec*.

Page, Sébastien, and Robert A. Panariello, 2018, When Diversification Fails, *Financial Analysts Journal* 74, 19–32.

##### Supporting Tables and Figures

From: Exploring the 60/40 Portfolio Downside

The magnitude of losses expected from history, showing the top five drawdowns and time to recovery of this hypothetical portfolio is shown below:

*Table shows the Top 5 drawdowns of this hypothetical portfolio and time in months to recovery.*

Figure 15: Hypothetical 60/40 Return Distribution shows the hypothetical return distribution of past asset returns for a portfolio using the current 30% offshore and split 60/40 between equities and fixed income. Although past performance is not necessarily what will happen in the future, we can make a number of observations. The average nominal return is positive as equity markets have been kind trending upwards. The blue shape maps the density using kernel density estimators to give an empirical probability distribution. It is clear the return distribution has not been normal. We see a bimodal nature with a hump of mass with small negative returns and a peak around 2.

*Figure shows the monthly total return distribution of a hypothetical portfolio generated from 18 year monthly returns for SA and International Equity and Fixed Income. A fixed allocation between the assets of 60% Equity and 40% Fixed Income with 30% offshore was used. Offshore asset returns are in ZAR. Density outline (in black) is a KDE plot which does not assume a parametric distribution. A Normal distribution centred on the mean is shown in purple.*

##### From: Costly Hedging?

To give an illustration of the protective put strategy, Figure 15 depicts the payoff profile. A protective put option is created by buying the stock or an equity index, and buying the associated put options. The protective put is also known as a synthetic long call as its risk/reward profile is the same as that of a long call. The whole idea of a protective put is to hedge against adverse market movements and not to realise profits from strategy, making this strategy equivalent to an insurance policy, rather than a speculative investment. Protection kicks in when losses are below a certain strike level. The total return of the strategy is reduced by the cost of the put options and the price path of the underlying assets (in our case the S&P 500).