A Simple Solution for Tail Risk

FFTW’s Thomas Philips on an enhancement to mean-variance optimization

Philips is not alone in having to face this conundrum, but he has been original and innovative in doing so in his role as global head of investment risk and performance at FFTW, a New York-based, fixed-income-focused firm that manages $56 billion on behalf of clients around the world.

“While algorithms to optimize tail risk are known,” he notes, “they tend to be complex and are not easily implemented. In addition, they tend to rely on historical data for their inputs, as they are not readily adapted to include forecasts of risks.”

In the interview that follows with Nupur Pavan Bang (Nupur_Bang@isb.edu), senior researcher at the Centre for Investment at the Indian School of Business in Hyderabad, Philips discusses a simple enhancement to the mean-variance paradigm that allows for the inclusion of tail risk.

According to Philips, this approach is easily implemented and has proven itself in the management of a wide range of fixed-income portfolios. It can be solved in closed form and typically results in a 15% to 20% reduction in tail risk relative to a classical mean-variance solution, with no change in volatility.

Philips, who has a PhD in electronic and computer engineering from the University of Massachusetts, is also BNP Paribas Investment Partners’ regional head of investment risk and performance for North America. (Fischer Francis Trees and Watts has been an affiliate of BNP Paribas since 1999 and wholly owned by the Paris-based bank since 2006.) Philips has also worked at the IBM Thomas J. Watson Research Center, the IBM Retirement Fund, Rogers Casey and Associates, Paradigm Asset Management and OTA Asset Management. He has published more than 30 research papers and is credited with two patents. He won the first Bernstein/Fabozzi/Jacobs-Levy prize for his paper “Why do Valuation Ratios Forecast Long Run Equity Returns?” and the Graham and Dodd award for “Saving Social Security: A Better Approach.”

Philips elaborated on his methods, their academic grounding and practical implementation during an August visit to the Indian School of Business.

NUPUR BANG: Markowitz’s mean-variance optimization model is one of the most widely used models in the investment industry. How does your model differ from his?

THOMAS PHILIPS: To understand that, you have to understand what mean-variance optimization addresses and what it does not. Mean-variance optimization allows us to build portfolios which appeal to us in two dimensions. In particular, it gives us a simple and computationally tractable way to relate the expected return and the risk of a portfolio to the risk and returns of its underlying assets, and a reasonable way in which to think about a trade-off between the two.

The last step involves utility theory, but even without the use of utility theory, mean-variance optimization is very useful. Before Harry Markowitz developed it, there wasn’t such a clear-cut focus on risk and return. A great deal of work, particularly on risk, had been done by gamblers and insurers, but their work was disjointed from the investment literature. The notion of a common person trading off risk and return came to the forefront because of Markowitz. And he made it accessible to a wide range of people because he used simple, computable measures of return and risk.

If you look at assets whose returns are largely a function of their mean and their variance, such as stocks, mean-variance optimization is a pretty good way to think about the problem of portfolio construction. But for assets such as bonds and options, it is a poorer approximation. Our model addresses the issue of how one ought to trade off risk and return when the distribution of asset returns is more complex.

Can you explain this further? Why is it a poor approximation for bonds and options?

Largely because their returns tend to be very skewed.

Let’s think about how corporate bonds are created. Risky corporate cash flows are split between two classes of investors: bondholders and stockholders. Bondholders are offered a relatively low, but correspondingly safe, rate of return. Equity holders are residual claimants to corporate cash flows — they get paid a higher (but risky) return. In particular, they get paid only if there is money left over after bondholders have been paid.

As a consequence, stockholders pick up almost all the variability in the company’s earnings, while bondholders experience little variability in their cash flows. But if a company gets deeply distressed (think of Enron) and then is unable to pay its bondholders, they take a huge hit. So, most of the time, bonds have a steady return, but once in a while they suffer huge losses. In other words, the distribution of bond returns cannot be normal.

It is worth pointing out that bonds typically come with protective covenants which give bondholders possession of the machines in the factory or the desks in the office in the event that the corporation cannot pay them what they were promised. In principle, at least, the bondholder can take those machines and desks and sell them at an auction to recover some, but likely not all, of their investment. It is commonly assumed that the recovery rate is about 40%.

How do you deal with this in practice?

One approach is to simulate the behavior of bonds and options, or to sample their historical returns and then build a complex optimization around these samples. Unfortunately, if the simulation model is not good, or if the historical returns don’t cover bad times as well as good times, one can get silly results. Another approach is to model the distribution of returns more accurately using a mixture of stable distributions.

Regardless of which approach one chooses, the level of mathematical sophistication rapidly escalates, and one has to be careful not to get trapped in a mathematical quagmire. Typically, when thinking about investments, simple math works best.

We address this problem by modeling risk in a simple, sensible way that is intuitive for fixed-income investors. We are happy to settle for a model that isn’t exact but points us in the right direction and is analytically tractable. In the special case when all returns are Gaussian, our model returns the same results as a classic mean-variance optimization. In other words, it is a good approximation in difficult cases, and exact in easy cases.

How is it different from some of the other efforts by, say, Black and Litterman?

There actually is a point of connection between our model and the Black-Litterman model. The key insight that underlies Black-Litterman is that one can use results from general equilibrium to get a basic solution in closed form, and can then modify this basic solution in accordance with some further insights on the relative expected returns of a few assets.

Our model is similar in spirit. We start by solving a simple mean-variance optimization problem in closed form, and then gently modify this solution in accordance with some insights that we have about the tail risk of each asset or strategy. Basically, if an asset or a strategy has a lot of tail risk, we decrease its allocation, and if an asset has very little tail risk, we increase its allocation. But after all the adjustments we make, the variance of the portfolio remains the same. So both solutions start with a simple solution and then modify it a bit in accordance with some auxiliary insights.

You discuss coherent measures of risk. Can you shed some light on this?

The theory of coherent risk measures was developed by Artzner, Delbaen, Eber and Heath in the mid to late 1990s. It turns out that many popular measures of risk, such as VaR, have some undesirable properties. In particular, they don’t satisfy something called the diversification axiom. If you combine two risky portfolios, you expect that the overall risk of the combined portfolio will not exceed the sum of the risks of its constituents. But Artzner et al. showed that under some widely used risk measures, you could have two portfolios with zero risk in isolation, but positive risk when combined. In essence, diversification was creating risk!

Any reasonable risk measure should not have this flaw. They went on to define a set of axioms that any reasonable measure of risk should satisfy, and call a risk measure that satisfies these axioms a coherent measure of risk. Markowitz used variance as his preferred measure of risk because it was both intuitive and tractable. Unfortunately, it is not coherent. We use expected shortfall as our risk measure because it is coherent and easily computed.

You are an example of how academic research blends with practice. How do you actually use your model?

FFTW is a fixed-income house, and we manage portfolios against a variety of benchmarks. We start by replicating the benchmark, and then layer on a set of active alpha strategies to build a complete portfolio. The alpha strategies come from several alpha teams. There is a structured securities team that analyzes mortgages, a rates team that analyzes the shape and level of yield curves, a money market team that focuses on the short end of the yield curve, a sector rotation team that rotates allocations between sectors, an FX team that focuses on currencies, a quant team that builds quantitative strategies, and an EM team that focuses on emerging markets. We compute the risk profile of each team’s model portfolio and use our model to allocate risk among the various alpha teams.

Have you found this to be much better than the traditional portfolio?

Yes, but in a very specific way. It reduces tail risk by 10% to 20% while leaving portfolio variance unchanged.

Could your model be extended to stocks, real estate and options?

It is easily applied to any asset class. However, as a general rule, I’d suggest using the simplest model that captures the key aspects of the problem you working on. Most optimization models work well when returns are approximately normal. If there is non-normality involved, tail risks get amplified, and you ought to use something like our model.

It is widely believed that asset allocation is 70% to 80% of the job and accounts for 70% to 80% of the returns that any investor gets. Stock selection, or selection of bonds/options or any other assets, accounts for just about 20%. Your model is a big step in deciding what an asset allocation should be in a portfolio. What are your general views on asset allocation?

This is an interesting question, and it is often misunderstood. There is a very nice paper by Roger Ibbotson and Paul Kaplan that appeared in the Financial Analysts Journal a few years ago and answers the most common variants of this question. If you are asking what fraction of the variability of your portfolio over time is explained by its asset allocation, the answer is about 90%. If, on the other hand, you ask what fraction of the cross sectional variation in return across funds is explained by asset allocation, the answer is about 40%. If you ask what fraction of your total return is explained by asset allocation, the answer is 100%.

I don’t think asset allocation has to be hugely subjective. But I do think that a few simple rules of thumb are very useful. You absolutely ought to diversify globally. And you ought to hold a wide range of assets. You won’t go too far wrong by having half your money in stocks and the other half in bonds, half domestically and half internationally. Is this perfect? No. Is it a reasonable starting point? Yes. It is even better if the bonds are indexed for inflation.

If you know your utility function, or if you can build good estimators of expected return, you could do better. But most investors don’t know what their utility function is. The only utility function that makes intuitive sense to me is the log utility function, because it corresponds to maximizing target wealth. Jarrod Wilcox had a paper in the Journal of Portfolio Management some years ago on an approach to maximizing return while preserving capital. In essence, he invested his discretionary wealth log-optimally and kept the remainder in cash equivalents. It’s a very clever idea.

What is your view on risks pertaining to countries? In the past two to three years, we have seen economies default which we never thought could go down. What are your thoughts generally on deleveraging and defaults by such economies.

Any country can get into trouble. A number of things went wrong in Europe and the U.S. in 2008, but they could well have happened anywhere. The deleveraging process is not easy, and it takes time. But I believe Europe will recover from the mess it is in. Policy makers are finally getting their act together, and pro-growth policies will soon start to take root.

What about India? Last year, the rupee depreciated by almost 25%, and “India shining” seems to have become a thing of the past, with uncertainties in policies and politics coming in the way of performance. What is your take?

I think you are being overly pessimistic. India is not a complete disaster that is falling apart, and it never was a perfect country that was headed straight up. It has always been something in between. There are (and were) pockets of innovation and pockets of stagnation. No one expected the IT industry to spring up in India. And I can see that happening again with pharmaceuticals. So, both good and bad things are happening, but I believe that India’s problems (like most problems) are fixable.

What is your view on the state of research in the field of finance and how disconnected it may be from the real world.

I always tell young people who are starting out on a career in research that the world offers them incredibly interesting problems to solve. Take a look at the world — don’t just read journals. The other thing that I stress is the need to be interdisciplinary. I am always shocked by the number of solutions to problems I face that come from other fields. For example, at FFTW we monitor portfolios using ideas from statistical process control, and we estimate volatilities using ideas from digital signal processing. It is also a good idea to study history. As the old saying goes, all that’s new under the sun is what you didn’t learn in your history class.