**"yield curve"**

Our review of hedge fund trading strategies continues with a discussion of yield-curve arbitrage (YCA), a form of fixed income arbitrage. I have previously written about the yield curve, convexity, and duration.

Our review of hedge fund trading strategies continues with a discussion of** yield-curve arbitrage (YCA)**, a form of fixed income arbitrage. I have previously written about the yield curve, convexity, and duration. Recall that for bonds not offering embedded features (such as puts and calls), a bond’s price and the interest yield move in contrary directions, giving an inverse association involving duration and yield. Higher yields mean shorter durations. The

*$duration**of a bond is product of the duration and the price (value); with units of*

**, it reflects duration changes in dollars rather than in percentages.**

*dollar-years*A parallel shift in a yield curve occurs when the yield on all maturities change by the same amount. More likely are changes in which the spread between short and long maturities increase (steepen) or decrease (flatten). A ** dumbbell** portfolio is loaded up on bonds at the short and long ends of the yield curve; conversely, a

**strategy involves the purchase of intermediate-maturity bonds. Yield-curve arbitrage is a trading strategy in which a trader exploits relative mispricings along the yield curve due to high institutional demand for selected maturities, among other reasons.**

*bullet*A well-known form of YCA is the so-called ** butterfly** trade: long dumbbells (the “

**” of the butterfly) and short bullets (the butterfly’s “**

*wings***) in a net-zero $duration spread trade. For example, you might set up a portfolio in which you are long 4-year and 8-year maturities, and short 6-year maturities. Small parallel moves in the yield curve would have little effect on this portfolio, since it has a net $duration of zero. However, large parallel moves in either direction will guarantee a positive return due to the positive convexity (yield vs. price) of the portfolio – in effect, one expects greater convexity in the wings than in the body. That sounds good in theory; in practice, yield curves usually experience complex movement patterns that can have an unexpected affect on the outcome of a butterfly trade.**

*body’*There are four popular types of butterfly trades:

– No cash is needed up front, since the cost of the long positions is offset by the proceeds from the short sale. Suitable prime brokerage structures are available such that the long position acts as collateral for the short position, so that zero cash flow is required initially. This strategy benefits from a flattening of the yield curve, because most of the $duration is in the wings of the butterfly.*Cash and $duration neutral weighting*– The trade is structured such that each wing of the butterfly has equal $duration. This strategy benefits from small changes to the yield curve, because the body is less convex than the wings. The position profits from a steepening of the yield curve. Note that this trade is not cash neutral, so return must exceed the cost of carry.*Fifty-fifty weighting regression*– A sophisticated trade in which the linear regression measuring the spread between the short wing and the body is regressed against the spread between the body and the long wing. The more-volatile short wing is more likely to move away from the body than is the long wing. So, say for example we determine a regression coefficient of 0.5; it means that a 20 basis-point change in the short wing spread would imply a 10 basis-point move in the long wing spread. Since most of the $duration is in the long wing of the spread, the strategy benefits from a flattening of the yield curve.*Regression weighting*– The relative maturities of the three components (short wing, body, and long wing) are used as the weighting of each component. The results of this strategy are very similar to the regression weighting scenario, except that the weighting factor will generally be higher than the regression coefficient.*Maturity weighting*

There are calculated risk measures that can be used by traders to determine whether the spread on each of the butterfly strategies is attractive and invites investment. Advanced readers can look up the model developed by Nelson and Siegel[1] to see how to partially hedge the risk exposures of different butterfly spreads.

Next time, we’ll continue our survey of fixed income arbitrage by taking a close look at ** basis trading**.

[1] Nelson, C.R., and Siegel A.F., (1987) “Parsimonious Modeling of Yield Curves”

*, Journal of Business,*

*60 (4), , p.473-489.*

## Yield Curve – Theories

*by Eric Bank on May 11, 2011 · No Comments · in Documenting the Search for Alpha*

Recall that yield curves (also known as the term structure of interest rates) plot debt maturities (the independent variable) against interest rates (the dependent variable).

Recall that yield curves (also known as the ** term structure of interest rates**) plot

**(the independent variable) against**

*debt maturities***(the dependent variable). Debt maturities indicate the length of the borrowing period for a debt instrument. We spoke last time on how a yield curve is shaped; today we’ll look at a few theories that attempt to explain yield curve behavior.**

*interest rates***Pure Expectations Theory (PET)**

In this theory, it is assumed that any maturity of debt can substitute for any other through the miracle of compounding. For instance, if you have a view as to what the one-year interest rate will be one year from now (the ** forward rate**), then you can determine the current two-year interest rate as the compounded sum of the current one-year rate and the one-year forward.

This generalizes to the ** geometric mean **of short-term yields as the determinant of long-term yields. A geometric mean differs from an average in the it is calculated by taking the

*n*th root of the product of

*n*terms. For example, if you have two terms, say 4 and 16, the average is of course 10, but the geometric mean is the 2

^{nd}root – the square root – of their product (64), or 8.

The Pure Expectations Theory accounts for the fact that yields tend to change together over time, but doesn’t explain the fine details of the shape of the yield curve. It posits that forward rates are perfect predictors of future rates, which they are not. It thus ignores interest rate risk and also reinvestment risk. The latter is the risk that one cannot reinvest interest payments at an expected rate. The theory also assumes that the ability to arbitrage among different maturity bonds is minimal.

**Liquidity Preference Theory (LPT)**

This is a variant of the Pure Expectations Theory. It basically adds a premium to the PET-calculated yield for long-term debt to account for investor preference for short-term bonds over long-term ones. This premium is called the term premium or the liquidity premium. It acknowledges the risks involved in holding long-term debt, which is more likely to experience catastrophic events and price uncertainty than is short-term debt. A second premium is also included in LPT, for default risk, which is more likely when holding a bond for a long period of time, once again due to uncertainty.

**Market Segmentation Theory (MST)**

This theory acknowledges that different maturities of debt cannot be substituted for each other. This results in separate demand-supply relationships for short-term and long-term debt. Since investors (assumed to be risk-adverse) prefer the less risky short-term maturities, the demand for short-term debt is higher than that for long-term debt, and thus prices of the former are higher, driving down their yields. This helps to explain the normal shape of the yield curve, but not the fact that long and short term rates tend to change in unison, since they are supposed to be two separate and independent markets.

**Preferred Habitat Theory (PHT)**

Preferred Habitat Theory is an extension of MST which posits maturity preferences, or habitats, for debt investors: some investors like 3-year bonds, some prefer 6-year maturities, etc. If you want to sell an investor a bond outside the investor’s preferred investment horizon, you must offer the investor a premium. Since it is assumed that more investors have short-term habitats, it explains the higher yields on long-term debt, and is consistent with the tendency of short- and long-term debt yield curve segments to retain their shape when overall yields change.

Now that we have a good understanding of yields and the yield curve, we can resume our review of hedge fund strategies. Next time: * fixed-income arbitrage*.

## Yield Curve – Structure

*by Eric Bank on May 2, 2011 · No Comments · in Documenting the Search for Alpha*

Yield curves (also known as the term structure of interest rates) plot debt maturities (the independent variable) against interest rates (the dependent variable).

Loyal readers know that we are currently surveying hedge fund strategies. As we pivot from equity strategies to fixed-income arbitrage, we will first take a short “time out” to learn about ** yield curves**. In this article we’ll discuss yield curve shapes; next time out we will explore the theories that attempt to explain yield curve behavior.

Yield curves (also known as the ** term structure of interest rates**) plot

**(the independent variable) against**

*debt maturities***(the dependent variable). Debt maturities indicate the length of the borrowing period for a debt instrument. The interest rate associated with a given borrowing period is a point on the yield curve. Curve segments between actual maturities are interpolated. Yield curves are specific for a given currency – the yield curves for U.S. dollar-denominated Treasury bonds will (almost always) differ from the analogous chart of U.K. pound-denominated gilts.**

*interest rates*Debt yield is the overall rate of return on a debt instrument. Debt that is “locked up”, as in a certificate of deposit, will usually offer a higher yield than an on-demand savings account, due to the higher certainty of holding the former to maturity. Yield curves normally ascend with time, but the *rate of increase *diminishes with increased time. The fancy phrase for this is an ** asymptotically upward slope**. It reflects the fact that it is riskier to hold longer maturities, because it’s harder to predict distant future interest rates than it is to predict near term interest rate. This increased uncertainty usually demands a

*risk premium*, i.e. a higher interest rate (known as a

**). This makes the most sense if investors anticipate a period rising short-term interest rates – current investors willing to tie up their money today must be compensated for forgoing higher interest rates tomorrow. But even if rates are not forecast to rise, the liquidity spread tends to give longer maturities higher yields.**

*liquidity spread*An ** inverted yield curve** is, as you might suspect, one in which short term rates are higher than long term ones. Inverted curves indicate an anticipated drop in interest rates over time. This is not considered a good thing – inverted yield curves are often associated with strong recessions and depressions. Forecasts of a weak economy motivate long-term investors to agree to lower yields, on the premise that yields may go even lower. On the plus side, inverted yield curves indicate a belief in low future inflation. However, in an economic panic, a

**may increase demand for, and thus lower the yield of, long-term government bonds.**

*flight to quality*Currency is a prime determinant of yield curve shape. Another determinant is the type of debt instrument: government bonds, bank debt, corporate bonds, and asset-backed securities. The latter three also vary by the creditworthiness of the issuer – debt with ratings of Aa/AA and above is less risky than lower-rated debt, and hence demands less of a premium over the interest rates for government debt. The ** London Interbank Offered Rate (LIBOR)** is a benchmark rate reflecting the interest rates top London banks charge each other for unsecured funds. LIBOR is important in the swap market because it is used to peg the floating rate leg of an interest rate swap. For this reason, the LIBOR yield curve is often referred to as the

**.**

*swap curve*As we noted, the normal yield curve has a positive slope, reflecting investor sentiment that economies will grow over time and that inflation rates will rise accordingly. Central banks, such as the U.S. Federal Reserve, control a country’s money supply in order to either keep a lid on inflation (by tightening the money supply, i.e. raising short term interest rates) or fight recession and deflation (by increasing liquidity and reserves in credit markets, i.e. lowering short term rates and/or printing more money).

Historically, the 20-year Treasury bond has yielded a ** spread** of two additional interest rate points above the three month Treasury bill. However, events can cause yield curves to steepen (larger spreads) or flatten (smaller spreads). Recently, the spread between 2-year and 10-year Treasuries widened to record-setting amounts approaching 3%. Sometimes, a yield curve is bicameral, or humped, meaning that medium-term yields exceed long- and short-term ones. This can occur when short term volatility is expected to be more significant than that in the long term. Flat yield curves can predicts steady interest rates or reflect general uncertainty about the economic future.

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