**"hedge fund strategies"**

## 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.

## Arbitrage Pricing Theory – Part One

*by Eric Bank on February 5, 2011 · No Comments · in Documenting the Search for Alpha*

Arbitrage Pricing Theory (APT) is a multi-factor model conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor.

Last time out, I promised we would look at the risk premia associated with various hedge fund strategies. But first I’d like to finish the discussion of pricing models that we began with our look at the **Capital Asset Pricing Model**.

**Arbitrage Pricing Theory (APT)** is a* multi-factor model *conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own *betas* to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor. Here are the equations:

Suppose that asset returns are driven by a few (*i*) common systematic factors plus non-systematic noise:

r_{i}= E(r_{i}) +b_{i}_{1} F_{1} + b_{i}_{2} F_{2} +· · ·+b_{in} F_{n} + ε_{i}_{ } for i = 1, 2, . . .n where:

- E(r
_{i})*i* - F
_{1}, . . ., F_{n}are the latest data on common systematic factors driving all asset returns - b
_{in}gives how sensitive the return on asset i with respect to news on the n-th factor (*factor loading*) - ε
_{i}is the idiosyncratic noise component in asset i’s return that is unrelated to other asset returns; it has a mean of zero

APT claims that for an arbitrary asset, its expected return depends only on its *factor exposure*:

If we define RP_{i} = the *risk premium *on factor i = the return from factor i above the risk-free rate = (rF_{n}−R_{f}), then the **APT equation** is:

E(r_{i}) = R_{f} +b_{i1} RP_{1}+. . .+b_{in} RP_{n} where

- RP
_{n}is the premium on factor n - b
_{in}is asset i’s loading of factor n - R
_{f}is the risk-free rate

So you can see, the expected return of an asset *i *is a linear function of the assets’ betas to the *n* factors. This equation assumes ideal markets and no surplus of the number of factors above the number of assets.

In Part Two, we’ll explore the assumptions and implications of the APT equation.

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