Bond Portfolio Investing and Risk Management
POSITIONING FIXED INCOME PORTFOLIOS FOR ROBUST RETURNS AFTER THE FINANCIAL CRISISBy VINEER BHANSALI
The McGraw-Hill Companies, Inc.
Copyright © 2010 The McGraw-Hill Companies, Inc.
All right reserved.
ISBN: 978-0-07-162370-4
Contents
Chapter One
Risk and Total Return
“There is nothing new under the sun but there are lots of old things we don’t know.”
—Ambrose Bierce, The Devil’s Dictionary, U.S. author and satirist, 1842–1914
The financial world changed in 2008. Many brokers become banks, and a large portion of the global financial system is owned or controlled by governments. Many fixed income markets that were “rates” markets, such as municipals, have become “credit” markets, and many credit markets have disappeared into a morass of defaults. What is an author supposed to do to relay principles of fixed income when the best he can do is to take a snapshot of a path-dependent process that is transmogrifying every day? Consider the yield on Treasury bills:
U.S. One-Month Bill Rate Negative for First Time Since December 2009-03-26 14:04:13.765 GMT
By Dave Liedtka, March 26 (Bloomberg)—Treasury one-month bill rates were negative for the first time since Dec. 26. The rate on the one-month bill was negative 0.0152 percent in New York, compared with 0.03 percent yesterday.
The only reason a person would accept a negative return and part with his or her money would be if the holding of the asset conferred some risk- mitigating benefit to the holder. In the case of the T-bill with negative yields, this benefit was the protection of capital. Investors were so scared of not getting their principal back that they were willing to give up return on their principal (actually they were willing to pay the federal government to keep their money safe).
The reason I start with this example is because return cannot be separated from risk. Unless a fixed income investor understands the inherent risks of an investment, it makes little difference where the return is coming from. To understand risks, we build models. Models are simply analytical tools that make sense, and they should not be confused with mathematical symbols or computational ability.
Model builders have lots of choices. What differentiates a good model from a bad model? My view is that it is the relevance to the markets and the ability to be robust to structural changes. A Wall Street trading desk that intermediates risk between two different counterparties uses its models as an inventory management system for important risks that do not impact the bottom line for short time intervals. So the models can be relatively simple and coarse. At the same time, a proprietary desk for the same dealer requires a more sophisticated set of models, especially if derivatives are involved, to ensure that there is no mispricing between similar or fungible assets. An asset manager who holds securities for longer time horizons requires even more truth in the models used because most of the reward for holding risks is through the risk premium realized. Valuation of risk premia, which is ignorable for a short-term trading desk, is the key issue for longer-term holders of risk. So, while a short-term trading desk can get away with risk-neutral valuation, another way of saying that the market is efficient in the short term, longer-term holders of securities and risk can hardly exploit short-term mispricing. For investors, the class that covers most of us, it is more important to be able to position portfolios to take advantage of risk premia than to arbitrage short-term mispricing.
To understand risk premia, we need to understand risks. A convenient way to describe the risks of securities is by using risk factors. For any security, we can postulate that the return is proportional to the return on some factors, and the exposure of the security to those factors, plus some idiosyncratic return. In other words, let us assume that we can write the returns of a security ri as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
where the N variables labeled fi … fN are the return on the fundamental variables we call factors that can influence the price of the security. Typically, we also impose the condition that the expected value of the noise term is zero, that is, E (εi) = 0, and that the factor movements are orthogonal to the noise term.
Now suppose that the mean return on the asset is [bar.ri]. Then the excess return is ri – [bar.ri] and can be written in terms of the excess factor returns:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
As long as the factor set is complete, this equation holds true for all assets. In this chapter we will identify and explore the factors that are relevant to the management of fixed income portfolios. Since the excess returns on securities can be expected to be proportional to the excess returns on the factors, the risk on securities is also linked to the risk on the factors.
Indeed, with the introduction of new types of securities, many exotic and really invented in the last decade, traditional risk factors are far from sufficient. This demonstrates the limitation of the factor approach. For such securities, current technological expertise does not allow much more than a rudimentary valuation of the security in various hypothetical but reasonable scenarios.
The power of the risk-based approach becomes apparent when we apply it to portfolio construction from the ground up. Suppose that we were to forecast that equity volatility over the next five years (our investment horizon) would average 20 percent a year (the long-term average or close to it). An equity exposure of 60 percent in our portfolio would translate this to a 12 percent volatility (0.60 times 20 percent) from the equity risk factor. This means an approximately 5 percent chance, of a drawdown of 1.6 times the volatility, that is, a 1 in 20 chance of a drawdown of more than 19.2 percent. Clearly, having this estimate as a rough starting point allows us to scale the big bets properly. Too frequently investors get focused on what to buy and not on how much risk to take. Similarly, for interest-rate risk, if we forecast a volatility of 100 basis points per year, then a 5-year-duration portfolio leads to approximately 5 percent volatility per year and a 1 in 20 chance of a drawdown of more than 8 percent over a year. However, if yields are high enough, the coupon income might subsidize the risks from negative mark-to-market, that is, the embedded carry can smoothen price-based return volatility. This observation highlights something that we will discuss in detail later—that carry, or structural return, is an integral part of robust portfolios.
Fixed Income Risk Factors
For fixed income securities, risk measurement is fundamentally a more complex task than it is for securities such as equities. There are a lot more moving parts. To name a few:
Shifts of the yield curve lead to duration and convexity risk.
Yield curve reshaping leads to what is called curve risk.
Various kinds of spreads can change without the yield curve changing. These result in spread durations.
* Mortgages
* Corporates
* Municipals
* Emerging
* Treasury Inflation Protected Securities (TIPS)
* Converts
Currency-rate movements result in exchange-rate risk.
Volatility and prepayment risk result in negative convexity (especially in mortgage-related securities).
Liquidity risk creates additional spread risk and possibly tail risk (to be described later).
The relevance of measuring these risks carefully is not simply for risk management and control but also for active alpha generation. Sustainable alpha is generated from exploiting attractive sources of risk premia. Risk management is simply the other side of the coin—it means systematically managing the risks from the sources of risk premia.
Perhaps the most important idea is that by managing investment portfolios using the factor approach, we can achieve dual objectives of efficient risk management and alpha generation. If we can match the factor exposures of a portfolio using cheap securities and derivatives (if allowed), then there is a built-in bias toward outperformance. Many active bond funds have almost as many independent securities as are in their indices (such as the Barclays/Lehman Brothers U.S. Aggregate), but only a few hundred nonmortgage securities overlap (the mortgage pools make up the bulk of the line items in the portfolio as opposed to generic mortgage pools in the index). But the risk exposures, as measured by the risk factors described earlier, are very close to the index risk factors. Such a portfolio replicates the index risk factors but has built-in “alpha” from not holding each index security. Index securities typically trade richer due to holding by passive indexers who are required to purchase these securities in order to minimize tracking error to the indices.
Of course, in and of itself the reduction in overlap is not a sufficient objective. We have to make sure that the reduction in overlap actually improves the portfolio’s risk-return characteristics. In the example portfolios we mention, the reduction of idiosyncratic risk requires a larger holding of common corporate bonds, nonagency mortgages, and other credit-sensitive securities. This makes sense because bonds subject to default risk carry idiosyncratic risks, which is harder to justify using just factor exposures.
Different Ways of Measuring Risk
There are a number of ways that the risk statistics, once computed, can be used for analysis of the risk-return potential in a portfolio. The first one is simply stress testing or scenario analysis. We can take each factor that can affect the value of a security and move it by some large magnitude. The impact of the factor shock on the percentage change of the security’s price is the factor duration. For example, if we change the level of the yield curve by a parallel shift, the resulting impact (in percentage) on the price of a bond is simply the interest-rate duration relative to the Treasury curve. We could proxy the change in the yield curve by taking one or many points as reference.
However, this methodology does not say anything about the possibility that many of the risk scenarios can be realized simultaneously, that is, that the yield curve shifts up and flattens simultaneously. The approach also does not use any input on the probabilities of the particular scenarios. To tie in the correlations between the simultaneous movement of risk factors, we need to estimate a covariance matrix of the factors, either historically or ex ante, and compute the total risk as one number. The value of having one risk number is that different portfolios with different mixes can be compared. The shortcoming of this approach is that whenever disparate sources of risk are aggregated, there is a loss of information—aggregation done wrongly can lose more relevant information than the gain in simplicity from having one number for stating risk.
Finally, instead of estimating the total risk using a covariance matrix, one can estimate the risk by running actual simulations. You simply take the change in the factors over a predefined interval and see how the change affects the portfolio value.
The “Big 4” Risk Factors for Active Fixed Income and Total Return
A yield curve theoretically has an infinite number of maturity points that can fluctuate. However, to capture most of the risks of the yield curve, we do not need to describe the movement of each and every point in the curve. A simple analysis, for example, the one originally proposed by Litterman and Scheinkman [42] using principal components analysis, shows that three dominant movements—yield-curve shifts, twists, and curvature—capture approximately 85 percent, 10 percent, and 5 percent of all the volatility in the yield curve. Of course, this is a backward-looking statistical conclusion that says nothing about the future movements of the yield curve. To forecast the future movements requires much more thought and needs to draw on macroeconomic conditions, technical conditions, flows, and so on. Much of this will be discussed in a later chapter. For now, note that a framework of stylized factors is sufficient for us to construct the foundations for risk-factor allocation.
Duration is the risk factor that captures the response of a portfolio to the parallel shifts of underlying yield curves. For instance, a typical intermediate-term bond index has a duration of approximately five years. Of course the question immediately arises, why would anyone take duration exposure? The answer lies in the fact that taking duration-factor risk is compensated in terms of excess risk-premium return. Since extending duration requires tying up money for a longer period and giving up access to it, the compensation is in terms of higher yields. Every source of excess return is compensation for some option that is sold to someone else. In the case of duration, the option that is sold is the ability to rebalance to higher yields if yields rise in the interim.
Curve duration is the the risk measure that captures the impact on the portfolio from a steepening of the yield curve by 100 basis points. To describe the steepening, we need to pick a point that remains fixed (the pivot). We can use the 10-year point on the U.S. yield curve as the pivot. The reason for this is simple: We are trying to describe independent movements of the yield curve in terms of three or so factors, so it makes sense to pick factors that capture individually most of a particular type of risk actually observed in the market and that are consistent with the intuition of practitioners. The 10-year point is the benchmark for most global bond markets; hence parallel shifts are best described using the 10-year point as a proxy. Then the steepening factor should be constructed so that the parallel shift is as independent of the steepening movement as possible.
Spread durations are the percentage change in the portfolio from a change in the spreads of the bonds, not changes in the levels or shape of the yield curve. In practice, to compute spread durations, we have to compute the option-adjusted spread of a bond, hold the yield curve fixed, shift the option- adjusted spread, and then recompute the new price. We will have much to say about what option-adjusted spread really means, how it is computed, and whether it is a good measure for valuation in later chapters.
Convexity, or the concept that duration does not remain unchanged as the yield curve moves around, is what makes fixed income different and interesting. There are different types of convexities that arise in the bond markets, but the most fundamental type is the one that arises from the fact that prices and yields are related through a nonlinear relationship. To compute the convexity of a bond portfolio, we first need to compute the convexity of individual bonds or their derivatives. To compute the convexity of a bond, we can use analytical methods, but in most cases we need to do the computation numerically. For securities such as mortgages, this is achieved by simulation first for a shifted yield curve and then recomputation with shifts from the already shifted yield curve. Ultimately, the output is something that we can call bull duration and bear duration, where bull duration is the duration for a 50 or 100 basis point parallel yield curve shift downward (lower yield), and bear duration is the mirror-image shift. To give an idea of the magnitudes, for a typical benchmark such as the Barclays/Lehman U.S. Aggregate Index, the baseline duration is approximately 4.5, and for a 50 basis point shift up, the duration increases by 0.20 year to 4.7. Similarly, for a 50 basis point shift down, the duration falls by 0.20 to 4.30. As we can see, this indicates that this index is negatively convex (a typical, positively convex Treasury bond would lose duration when yields rise and pick up duration when yields fall). Where is the negative convexity coming from? The answer is almost completely from mortgages. Note that almost 35 percent of total market value and 30 percent of interest-rate duration risk of the Barclays/Lehman Index is from mortgages. Since mortgages are negatively convex securities (owing to the embedded prepayment option), the index picks up the negative convexity.
Extra yield is the reward for incurring the negative convexity risk. Mortgages compensate the investor for the negative convexity via a higher yield than a comparable-duration Treasury bond. If time passes and markets come to a standstill, positive convexity is a waste. Flipping the argument on its head, if markets come to a standstill and only time passes, then the yield compensation an investor received for selling the prepayment option and incurring the negative convexity results in excess return.
Broadly speaking, the risks, and thus the returns, of a fixed income portfolio originate from the sum of things that change and the passage of time. Let’s explain what this means in some detail. When a portfolio is positioned to take advantage of changes in the exposure to some market factor, say, duration, the return is simply the duration times the change in yield. But, even if the market does not change, the portfolio has some excess positive or negative return simply because time passes.
(Continues…)
Excerpted from Bond Portfolio Investing and Risk Managementby VINEER BHANSALI Copyright © 2010 by The McGraw-Hill Companies, Inc.. Excerpted by permission of The McGraw-Hill Companies, Inc.. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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