White Papers

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This paper was based on a conference presentation at the Journal of Investment Management in 2014. I have long had the view that what we do for the love of the activity (i.e. hobbies), has a strong relevance to our professional lives, especially for investors. It allows us to think outside of the box and in both visualizing and solving problems with approaches that pay attention to the most important elements. I use my experience from ultra-running, flying, programming, physics and screenwriting to draw upon some lessons relevant for investing.

Why do people hedge? Who do others sell the same hedges? It is impossible to address these questions convincingly without paying careful attention to behavior. In this paper we address four aspects of tail hedging. First, having tail hedges makes people feel better, and it is hard to come up with a perfect model that can predict behavior. Thus it is hard to say with absolute certainty and objectively when such hedges are cheap or expensive. Second, it is possible to build models using a behavioral framework that can quantitatively justify the option skew in index options markets. Third, within a behavioral context it is possible for both hedgers and non-hedgers to co-exist without either one being irrational. Finally, tail hedging allows us to overcome time inconsistency in investment decision making, hence serves an important purpose in constructing robust and consistent portfolios.

Retirees are faced with the real world problem that they cannot repeat their experience multiple times and then choose one path. For different age cohorts, the risk of drawdown or loss determines how much risk they can take, and how they react to sharp drawdowns in the markets. In this paper we create a framework for the optimal amount cost for hedging tail risk that a retiree should be willing to bear either implicitly or explicitly.

We know the two key maxims of investing: don’t fight the market (i.e. be on the right side of the trend) and don’t pay too much to invest (i.e. control the negative carry in each investment). We find through empirical analysis that these two simple elements indeed bear out the test of time. Following a simple trend following strategy while minimizing or eliminating negative carry indeed is simple yet efficient way to deliver consistent performance relative to other alternatives. This simple approach seems to hold up regardless of eras and market environments.

Risk Management is a multi-faceted enterprise. Using the four levers of risk management: dynamic rebalancing, alternative risk factors, explicit tail hedging, and efficient management of liquidity are all important when thinking of the right mix of strategies for risk management. The best mix is found by optimizing the costs vs. benefits of these four levers, which are dynamic and ever changing.

Market return distributions in the modern era have looked more like a “two-humped” camel’s back than a “one-humped” camel. If we believe that this bimodality is likely to persist then many of our notions of how to construct portfolios and manage risks come under scrutiny. For example, the allocation to risky assets is decreased, exposure to momentum assets is increased, and the purchase of fat tails on both the left and right side becomes more attractive.

Risk Parity falls into the class of systematic investment strategies that start from the basic premise that it is easier and more practical to start from risk control as an objective rather than forecasting returns. This is a solid approach towards portfolio construction, but can be improved with active management. Paying attention to risk factor concentrations, the levels and expected returns of various markets, and to the possibility of a breakdown in historical correlations under stress are all relevant towards creating a superior version of passive risk parity.

By using a risk factor approach, we can peek at the hidden risks of risk parity strategies. We find that with a very simple two-factor model of equities and duration risk, we can efficiently understand where risk parity returns come from, and how to mitigate and control such risks more efficiently.

Portable alpha strategies take the alpha or non-systematic source of returns from one strategy, and apply to another underlying source of beta. We discuss how to control risk when this strategy comes under stress by deriving formulas and results for the optimal “stop-loss” rule.

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While most investors think of Tail Risk Hedging as a net cost to portfolio returns, our analysis shows that active Tail Risk Hedging is not only risk reducing but also return enhancing. How can this be? First, Tail Risk Hedging allows, for a small premium, to stay in attractive risk positions; second, by monetizing and actively managing a portfolio of hedges, investors can potentially systematize their portfolio rebalancing, which allows them to stick to their plan of investing when risk assets have cheapened.

The short lived but attractive life of taxable municipal bonds known as BABs showed an interesting dynamic in the transfer of investment value from one class of investors to another class of investors. BABs also provided an important laboratory for the analysis of various risk factors that drive municipal bonds. The BAB program succeeded in lowering the cost of funding for state and local governments with BAB issuers obtaining finance 54 basis points lower, on average, compared to issuing regular municipal bonds. For institutional investors, BAB issue yields were 116 basis points higher than comparable Treasuries and 88 basis points higher than comparable highly rated corporate bonds. For individual investors, BABs had lower yields than regular municipal bonds. Thus, on average the Federal government subsidy disadvantaged individual U.S. taxpayers, who were the main holders of municipal bonds, and benefited new entrants in the municipal bond market.

How Do Policy Makers Impact Markets? Traditional pricing and risk management principles have to be reconsidered in the light of the involvement of large public and private participants in the securities markets. We explore the impact of the actions of participants on the building blocks of asset pricing models as well as the emergence of the impact of the policy factor on asset pricing and risk measurement. This discussion re-emphasizes the need for simple principles for asset selection and portfolio construction.

The “risk-management paradigm of monetary policy” that has been followed by the recent Fed and many other global Central Banks can be modeled by adding an option like term to the Taylor rule as a guide by the Fed in setting policy rates. Since the evolution of short rates is what determines longer term rates and the whole yield curve, we can run simulations to show how different assumptions for the risk management term influences the shape of the yield curve.

How should we model swap spreads when markets change their behavior? By combining a simple two factor, analytically solvable model for the risk-free government yield curve with a hazard model for swaps, we can derive closed form solutions for the behavior of swap spreads in both quiet and stressed markets.

In physics, even when one or more critical parameters change slowly, macroscopic systems can undergo sharp changes (e.g. conversion of water to vapor as temperature or pressure rises). This process of phase transitions has close analogies in financial markets, with factors such as liquidity and leverage driving sharp changes and critical market events.

Under the TALF (Term Asset-Backed Securities Loan Facility) program that was introduced during the financial crisis, an investor had the ability to borrow via term loans against eligible TALF assets as collateral. Due to the non-recourse feature of these loans, the borrower in effect possessed a put option for each individual asset in the portfolio, with the maximum amount at risk to the borrower equal to the haircut. We estimate the value of the put under rough but reasonable assumptions, and also show that the portfolio of puts may be significantly more valuable than a hypothetical put on the whole portfolio.

By utilizing a three factor model calibrated to the prices of index credit default swaps and tranches, we can extract some very interesting macroscopic and microscopic features of financial markets. We find that the three factor model carries a very powerful and significant amount of information on how systemic risk is priced in the credit markets.

What is Tail Risk Management and how is it important for portfolio managers and risk control? This paper lays out a framework for thinking about the necessity and implementation details of tail risk hedging.

Are tax-exempt municipal bonds completely tax free? Due to certain taxation rules that are not widely understood by the public, we have to be careful about making this assumption when municipal bonds are close to their “de minimis” threshold, which, amongst other things, depends on the maturity of the bond, the time left to maturity, the issuance coupon and current yields. Close to this threshold, municipal bond taxation characteristics can change significantly, and create possible investment value.

Has financial modeling and investment practice moved so far that it has lost touch with the fundamentals? In this paper we look at rigorous financial models and build a framework to incorporate economic insights into the models.

Are the carry trade, where one borrows in a low yielding currency and invests in a higher yielding currency, and the level of volatility in the currency markets related? Certainly it seems so, since carry trades do well when volatility is low and not so well when volatility is high. We discuss an analytical foundation on why this should be so, by drawing a connection between the forward exchange rate that depends on interest rate differentials and the level of implied volatility, which can be locally used to hedge the risk of the forward position. This approach yields insights on when the carry trade is attractive relative to the level of currency volatility.

This article deals with the problem of optimal allocation of capital to corporate bonds in fixed income portfolios when there is the possibility of correlated defaults. Using a multivariate normal Copula function for the joint default probabilities we show that retaining the first few moments of the portfolio default loss distribution gives an extremely good approximation to the full solution of the asset allocation problem. We provide detailed results on the convergence of the moment expansion and explore how the optimal portfolio allocation depends on recovery fractions, level of diversification and investment time horizon. Numerous numerical illustrations exhibit the results for simple portfolios and utility functions.

How does the default probability of a corporate bond change as the underlying company changes its dividend payout policy? In this article we use the Merton model to explore this problem to come up with a dividend spread risk measure for corporate bonds.

The probability distribution for the relative return of a portfolio constructed from a subset n of the assets from a benchmark, consisting of N assets whose returns are multivariate normal, is completely characterized by its tracking error. However, if the benchmark asset returns are not multivariate normal then higher moments of the probability distribution for the portfolio’s relative return are not related to its tracking error. We discuss the convergence of generalized tracking error measures as the size of the subset of benchmark assets increases.

This article deals with the problem of optimal allocation of capital to corporate bonds in fixed income portfolios when there is the possibility of correlated defaults. Under fairly general assumptions for the distribution of the total net assets of a set of firms we show that retaining the first few moments of the portfolio default loss distribution gives an extremely good approximation to the full solution of the asset allocation problem.

When market crises happen, correlations rise in absolute value, and drastically change the risks of portfolios. We discuss a systematic methodology to forecast portfolio risk when correlations are shocked to extreme values. We achieve this by giving the forecaster control over the level of confidence in forecasting such correlations, and then taking the correlation matrix to its logical extreme value without destroying mathematical consistency.