The Application of Numeraire Portfolio in Pensions

The complexities of pension fund management and maximizing retiree returns are compounded by “enormous risks,” observes Eckhard Platen, professor of quantitative finance at the University of Technology, Sydney. “There is a need of a little bit of academic direction to do something about it. There is a way to do it. Even developed countries need to wake up and say, ‘now something has to be done.'”

Noting that classical finance theory has not filled all the gaps, Platen brings to the discussion a “Benchmark Approach” that, in a recent interview, he put in the context of a numeraire portfolio and how it could be applied to the practical issues and concerns surrounding pension schemes. At UTS Sydney since 1997 –as a joint appointment of its School of Finance and Economics and School of Mathematical Sciences — Platen has been contributing to efforts to improve the pension system in Australia.

With a Ph.D. in Mathematics from the Technical University in Dresden and a Dr.Sc. from the Academy of Sciences in Berlin, where he headed the Sector of Stochastics at the Weierstrass Institute, Platen has written more than 150 research papers in quantitative finance and applied mathematics, and his books include “A Benchmark Approach to Quantitative Finance” (Springer, 2006).

This interview conducted by Nupur Pavan Bang, senior researcher, Centre for Investment at the Indian School of Business in Hyderabad, covers the evolution of Dr. Platen’s views on diversification principles and risk neutrality, the numeraire portfolio and practical approaches to pension investing.

You do not believe in the classical principle of no-arbitrage. That is the most
fundamental principle of asset pricing theories in financial literature.

In the early 1990s, when I started working with leading investment banks in Australia, I was confronted, in practice, with the classical paradigm called the no-arbitrage pricing theory. Over time, it became clear to me that this theory is too narrow; the assumptions made are too constraining, and we must have an alternative that is more appropriate, especially in the case of longer-term contracts. It just occurred to me that something is wrong. The classical notion of no-arbitrage restricts too much the modeling world. There is a less restrictive and economically very reasonable notion of arbitrage that one should use. In principle, we should be able to create financial contracts and hedge them in a way that is less expensive than what classical theory propounds. Over the years, I was able to demonstrate how this is indeed possible.

Are pension schemes, therefore, very long-term contracts where you think your theory can be applied?

Think of it this way: Governments, companies and individuals contribute to pension funds. The amount is very big for most of us. As soon as we start earning, we are told that this contribution is mandatory. The cumulative contribution over the years is very significant, because if you start earning at the age of 20, you will end up contributing for 40 to 45 years. Lots can be done with this money from the time when the contribution is made and the payoffs are really paid off.

What are the concerns with the traditional pension schemes? What went wrong?

It is an enormous challenge to gauge a developed market and an emerging market in the same light. To start with, the way pensions have developed, they have emerged as the core of the welfare system. Secondly, the defined benefit pension plans have failed in the countries of the West. Thirdly, large companies like General Motors and Ford are just pension-generating companies, nothing else. Fourth, governments organize pensions based on an average mortality.

The aging population is a concern in a lot of countries. Let’s say people live longer — say, 15 years longer on an average. The calculations completely go wrong. Also, some major industries of some countries or some economies, they may just decay or vanish. We have seen this many times. For example, if a pension scheme invests a big chunk of the money in, say, a mining group in Australia. It could happen that after 15 or 20 years, the mining action shifts to Africa. What do you do then? So it would be very good if a scheme is widely diversified, so that its performance does not depend on a single industry or nation.

There are many different types of contribution schemes nowadays. Doesn’t that solve the problem?

No. Look at the equity-based contribution schemes that are now very popular in the developed nations. They often force people to liquidate everything at retirement date and buy an annuity. If the market is down in a particular year, and equities form 40% of the pension, then that is enormous risk. I would say this has to be removed soon.

What can be done?

One thing that should not be done is setting up a defined benefit scheme. Because you can never guarantee payoffs at the end of 5, 10 or 30 years. But what one can do is pool all the knowledge and best practices to create a mutual scheme or a pension scheme. The scheme should be diversified in terms of age of the participants, and it should have a large base of participants. Then keep adjusting the payoffs from time to time, depending on changes in life expectancy and economic conditions. So we aim to get the highest possible payout in the least expensive way. The challenge is how to set up such a scheme and make it fair so that, in principle, it is fully transparent to everyone.

How can such a scheme be set up?

There are different contribution plans that give different choices at retirement. This has to be put on the table. That is, if one contributes a certain amount in cash, then something will be done with that cash. It will be invested using certain strategies that target the highest possible payout. When you retire, you will get a payoff stream, like that of a life annuity, in cash. There is an agreement in principle that you get payments only until you die.

This pension has to be fully sustainable, fair and, of course, globally diversified. Since the level of payouts cannot be fully guaranteed, it should be a targeted life annuity. The entitlement depends on how much one paid in — how many units of the life annuity one has purchased over time by contributions to the scheme. The assets and liabilities are matched regularly over time, using what I call real-world pricing. They are adjusted periodically, based on the latest mortality figures and model adjustments available.

In some countries — for example, India — the regulations do not allow for global diversification of the pension pool.

Yes, certain countries have certain regulations, and one can operate only under these regulations. I believe the policy in India is to very much keep pension funds inside the country. It may be too early now, but there will come a time when the authorities will realize the advantages of diversifying beyond the country.

Can you explain your approach in more detail?

I use the Benchmark Approach, a kind of more general framework that we have today in finance. It does not take anything away from existing classical theory. John Larry Kelly Jr., of Kelly criterion fame, published a paper, in 1956, founded on maximizing expected portfolio growth based on logarithmic utility and gambling contracts.

In July 1990, the Journal of Financial Economics,a mainstream finance journal, published “Numeraire Portfolio,” a paper by John Long Jr. This is a portfolio which, when taken as a “numeraire” or a benchmark, and some given portfolio is denominated in units of this benchmark, then the current benchmarked value of the portfolio is greater than or equal to its expected future benchmarked values. Using this insight, we can potentially get a fair benchmarked portfolio process, forming a so-called martingale, where the current benchmarked value is equal to the expected future benchmarked values. I am suggesting to search always for this least expensive portfolio to hedge future payoffs. When doing this, use the real-world pricing formula, assuming that there exists a numeraire portfolio — the benchmark — and the expectations are taken under the real-world probability measure that models future change.

Doesn’t most of the financial literature use a risk-neutral probability measure?

Yes. Almost 90% of the literature. But I don’t believe that the risk-neutral probability measure exists. In fact, I know that this measure does not exist when fitting long-term models. Parts of the industry see the problem too, and that is why the largest reinsurance company in the world is interested in what I found.

I can tell you that I am a non-equity premium puzzle person, because the modeling world that the Benchmark Approach provides is so rich that a high equity premium is not a puzzle at all. It’s like you want to force a classical risk-neutral model [to calculate a risk premium] onto something where, in principle, you should accept and use the observed risk premium, which is higher than the classical theory allows. It is just another indication that the classical theory is too narrow.

How does your theory compare with the classical theory?

While we don’t take anything away from the classical theory, we go into a richer modeling world by making a very simple assumption: that there exists a numeraire portfolio. The numeraire portfolio, when used as a benchmark, makes all benchmarked non-negative portfolios supermartingales — their current benchmarked value is greater than or equal to the expected future benchmarked values. The greater-than-or-equal sign indicates the crucial supermartingale property. Since this property holds for all non-negative benchmarked portfolios, one can say that the numeraire portfolio is the best portfolio in this sense. It performs so well that when used as a benchmark, it forces all non-negative portfolios in expectation down, besides those that are martingales. With the supermartingale property, and no extra assumptions, I can prove that this portfolio in the long run outperforms any portfolio. It’s a dream portfolio.

It is also the portfolio that maximizes the expected logarithmic utility. It is growth-optimal. It’s a portfolio that in the shortest time reaches a certain level. It is the portfolio that cannot be systematically outperformed in any time period by any other portfolio. All these properties are model-independent and, thus, very robust. In fact, even the Indian market, if you take a close look at it as an investment universe, has its own numeraire portfolio somewhere, extremely well-performing. The question is just to find and construct it.

How do you account for the down side?

If I look with the Benchmark Approach beyond the classical theory, then there are some classical arbitrage opportunities in this richer modeling world. However, these are strategies and portfolios where I have to allow, for certain periods, some probability to become negative. But I argue that it is not necessary to exclude those strategies. We should look only at non-negative portfolios, because, with a notion of reasonable economic sense, when the worth of market participants becomes negative, then we have to remove them from the market because they are bankrupt. We take limited liability into account, but there is no need to look at the negative portfolios and exclude arbitrage for these.

This supermartingale property is also in this respect very elegant and powerful. It provides the simple mathematical conclusion that any non-negative supermartingale that reaches zero will never get out of zero. In this sense, one cannot create out of zero capital some positive wealth with a non-negative portfolio. This type of arbitrage, called strong arbitrage, is then automatically excluded in the wider modeling world of the benchmark approach.

How do you construct the numeraire portfolio?

The numeraire portfolio is very diversified — the best diversified portfolio you can build. In principle, it is capturing the non-diversifiable risk of the market that follows from a theorem that I have. Of course, this clings very much to the classical theory. Harry Markowitz once told me that I should call my theorem the Diversification Theorem, which brings all the finance theorems and principles together.

Using this diversification theorem, I create, in its simplest application, an equally weighted index (EWI), equi-weighted over companies, industries and countries. This index has a higher growth rate and higher Sharpe ratio (as well as lower volatility) when compared to the index weighted by market capitalization. It is a better proxy for the numeraire portfolio than the market-cap-weighted index. It can be used directly in portfolio management, as a best performing portfolio, as a benchmark.

The larger the number of companies, the closer we get, in principle, to the numeraire portfolio. The fundamental Law of Large Numbers is at work here. The market needs to be well securitized, which is a very simple and easy condition. In a well-securitized market, with an increasing number of securities, the sequence of equally weighted indices is a sequence of approximate numeraire portfolios. This is something that is covered by the Naïve Diversification Theorem.

How does the numeraire portfolio work as an investment strategy?

The strategy of an EWI is to buy low and sell high, and if a market is always trending up, you get people only wanting to buy and not to sell. But this fund will sell in such a scenario. On the other hand, if the market crashes, this fund will buy. So it has a very stabilizing effect on the market. The systemic risk in the market, with this kind of portfolio on a macroeconomic scale, is reduced.

Then there are people who might say, “All this is good, but what about transaction costs?” At 40 or 80 or even 200 basis points, the performance, of course, goes down but this portfolio still performs better than the market-capitalization-weighted portfolio. The Sharpe ratio is still better for the proxy of the numeraire portfolio. It is a very robust, stable kind of situation.

How can the pension fund industry use the numeraire portfolio?

The pension fund must invest in a proxy of the numeraire portfolio. Not just that, but the numeraire portfolio also gives us a pricing rule. In the supermartingale world, we call it the fair price process. Let’s take a savings account; and also a proxy of the numeraire portfolio, a benchmark; and let’s benchmark the savings account. Over the long term we get an on-average downward-sloping fluctuating curve for the benchmarked savings account, because the numeraire portfolio is going up more in the long run than the savings account. So this is a self-financing portfolio. Financial planning tells you that when you are young you should invest in the equity market (the benchmark), and then in later years fixed income (the savings account).

Using the numeraire portfolio and the savings account, about which I just spoke, we can construct a self-financing hedging strategy, where according to some model, you invest almost everything into the numeraire portfolio when you are young and then slide over, in a precisely defined manner, to the savings account over time.

Does this portfolio take care of inflation? In a country like India, where inflation is on the higher side, people are worried about the time value of their money.

People like their pension payouts to be inflation-indexed. That is because the pension contract is very long-term and much can happen over that period. In my experience, interest rates are generally a percentage or two higher than inflation in most economies. It is very difficult over long time periods to get the interest modeled correctly. So why not take it out completely? What we do is assume that one unit of payment is equal to one unit of the saving account. All the payments that you contribute and get later on in your post-retirement stream are in the form of savings account units. In particular, the targeted payouts are units of the savings account. So when you purchase your life annuity, you get rid of the risk or uncertainties from the modeling problems coming from the short rate. My payout unit is the unit of the saving account; my basic instruments are the benchmark and the savings account.

Current actuarial methodologies focus primarily on modeling the interest rate evolution to value pension funds and life annuities. Several developed countries have moved to a zero-interest-rate regime. This creates problems for the growth of wealth when using classical actuarial methods. Using the benchmark approach and the proposed targeted pension, one can avoid several of the currently burning problems. In the design of new pension schemes, one can benefit in several ways.