Fixed Income Lesson 1

Par rate and the par curve

A par rate is the yield to maturity of a bond trading at par, so it equals the bond’s coupon rate. Par rates across maturities form the par curve, normally built from government or benchmark bonds. The par curve is observable but not directly usable for valuation, because each par rate is a single blended yield covering cash flows at many different dates — which is why spot rates are bootstrapped out of it.

Bootstrapping

Bootstrapping derives spot (zero-coupon) rates from the par curve, using the output of each step as the input to the next. For annual-pay bonds it starts with the one-year spot rate, which equals the one-year par rate because a one-year bond has only one cash flow. You then take the two-year par bond, discount its first coupon at that known one-year spot, and solve for the rate that makes the remaining cash flow price the bond at par — that is the two-year spot. Repeat for three years using the first two spots, and so on down the curve.

The swap rate curve

Swap rates across maturities form the swap rate curve, arguably the most commonly used interest rate curve, roughly reflecting the default risk of a commercial bank rated around A1/A+. Participants prefer it to the government bond yield curve for three reasons: it reflects the credit risk of commercial banks rather than governments; the swap market is unregulated, which makes swap rates comparable across countries while government curves additionally carry sovereign risk unique to each country; and the swap curve has quotes at many maturities whereas on-the-run government issues trade at only a small number. Wholesale banks, which manage interest rate risk with swap contracts, value their assets and liabilities off the swap curve. Retail banks use the government bond yield curve.

Swap spread

The swap spread is the amount by which the swap rate exceeds the yield of a government bond of the same maturity: swap spread = swap rate − Treasury yield. It measures bank-sector credit risk against the risk-free government rate at matching maturity. If the fixed rate on a one-year fixed-for-floating MRR swap is 0.57% and the one-year Treasury yields 0.11%, the one-year swap spread is 0.46%, or 46 basis points.

TED spread

The TED spread is the amount by which MRR exceeds the interest rate on short-term US government debt of the same maturity, usually three-month. The name combines the T in T-bill with ED, the ticker for the Eurodollar futures contract. Because T-bills are considered risk free while MRR reflects the risk of lending to commercial banks, the TED spread is read as an indication of credit and liquidity risk in the banking sector. A rising TED spread indicates that market participants believe banks are increasingly likely to default on loans and that risk-free T-bills are becoming more valuable in comparison. It captures risk in the banking system more accurately than the 10-year swap spread does.

MRR-OIS spread

OIS stands for overnight indexed swap and represents the interest rate on unsecured overnight lending between banks. The OIS rate roughly reflects the federal funds rate and includes minimal counterparty credit risk, because the exposure lasts only one night. The MRR-OIS spread, formerly the LIBOR-OIS spread, is the amount by which MRR — which includes some credit risk over a longer term — exceeds the OIS rate. Like the TED spread, it indicates the level of credit and liquidity risk in the banking system.

I-spread

The I-spread, or interpolated spread, for a credit-risky bond is the amount by which the yield on that bond exceeds the swap rate for the same maturity. It compensates the investor for the issuer’s credit and liquidity risk. Where the swap rate for a specific maturity is not available, the missing swap rate is estimated from the swap rate curve using linear interpolation — which is where the name comes from.

Z-spread

The Z-spread is the spread that, when added to each spot rate on the default-free spot curve, makes the present value of a bond’s cash flows equal to the bond’s market price. Unlike the other spreads it is measured over the entire spot rate curve rather than against a single benchmark yield. The term zero volatility refers to its assumption of zero interest rate volatility, and because options are meaningless without interest rate volatility, the Z-spread is not appropriate for valuing bonds with embedded options. If you ignore the embedded options and estimate a Z-spread anyway, the estimate comes out too wide, because it includes the cost of the embedded option — reflecting compensation for option risk as well as for credit and liquidity risk.

What makes a forward price change

If spot rates evolve exactly as the forward curve implied, the forward price will remain unchanged. Therefore a change in the forward price indicates that the future spot rates did not conform to the forward curve. When spot rates turn out to be lower than implied by the forward curve, the forward price increases; when they turn out higher, it decreases. A trader expecting lower future spot rates than the current forward rates imply would therefore purchase the forward contract to profit from its appreciation.

Riding the yield curve

The most straightforward strategy for a bond investor is maturity matching — purchasing bonds whose maturity equals the investment horizon. With an upward-sloping term structure, an investor seeking superior returns may instead ride the yield curve, also called rolling down the yield curve, purchasing bonds with maturities longer than the horizon. On an upward-sloping curve shorter maturity bonds have lower yields, so as the bond approaches maturity and rolls down the curve it is valued using successively lower yields and therefore at successively higher prices. If the yield curve remains unchanged over the investment horizon, riding the yield curve produces higher returns than simple maturity matching. The greater the difference between the forward rate and the spot rate, and the longer the maturity of the bond, the higher the total return.

Unbiased (pure) expectations theory

Under the unbiased expectations theory, also called the pure expectations theory, it is investors’ expectations that determine the shape of the interest rate term structure. Forward rates are solely a function of expected future spot rates, and every maturity strategy has the same expected return over a given investment horizon — an investor should earn the same return from a five-year bond as from a three-year bond followed by a two-year bond. In other words, long-term interest rates equal the mean of future expected short-term rates. The underlying principle is risk neutrality: investors do not demand a risk premium for maturity strategies that differ from their investment horizon.

Local expectations theory

Local expectations theory is similar to the unbiased expectations theory with one major difference: it preserves the risk-neutrality assumption only for short holding periods, so over longer periods risk premiums should exist. This implies that over short time periods every bond, even a long-maturity risky bond, should earn the risk-free rate. The theory can be shown not to hold, because the short-holding-period returns of long-maturity bonds are higher than the short-holding-period returns on short-maturity bonds, due to liquidity premiums and hedging concerns.

Liquidity preference theory

Liquidity preference theory addresses the shortcomings of the pure expectations theory by proposing that forward rates reflect investors’ expectations of future spot rates plus a liquidity premium compensating them for exposure to interest rate risk. That premium is positively related to maturity, so a 25-year bond should carry a larger liquidity premium than a five-year bond. Because forward rates include a premium, they are biased estimates of the market’s expectation of future rates. A positive-sloping curve may therefore indicate either that the market expects future rates to rise, or that rates are expected to remain constant or even fall but the addition of the liquidity premium produces a positive slope. A downward-sloping curve indicates steeply falling short-term rates. Liquidity premiums need not be constant over time and may be larger during periods of greater economic uncertainty, when risk aversion is higher.

Segmented markets theory

Under segmented markets theory, yields are not determined by liquidity premiums and expected spot rates. Rather, the shape of the yield curve is determined by the preferences of borrowers and lenders, which drives the balance between supply of and demand for loans of different maturities. It is called segmented markets because the yield at each maturity is determined independently of the yields at other maturities — each maturity is essentially unrelated to the others. The theory supposes that market participants deal only in securities of a particular maturity because they are prevented from operating at different maturities: pension plans and insurance companies, for example, primarily purchase long-maturity bonds for asset-liability matching reasons and are unlikely to participate in the market for short-term funds.

Preferred habitat theory

Preferred habitat theory also proposes that forward rates represent expected future spot rates plus a premium, but it does not support the view that this premium is directly related to maturity. Instead it suggests that an imbalance between the supply and demand for funds in a given maturity range induces lenders and borrowers to shift from their preferred habitats to one with the opposite imbalance. To entice them to do so, investors must be offered an incentive compensating for exposure to price and reinvestment rate risk in the less-than-preferred habitat: borrowers require cost savings, meaning lower yields, and lenders require a yield premium, meaning higher yields. Because premiums relate to supply and demand at various maturities rather than to maturity itself, a 10-year bond might carry a higher or lower risk premium than a 25-year bond — which is why preferred habitat theory can be used to explain almost any yield curve shape.

Effective duration and shaping risk

Effective duration measures price sensitivity to small parallel shifts in the yield curve. It is important to note that it is not an accurate measure of interest rate sensitivity to non-parallel shifts, and exposure to those is described by shaping risk — changes in portfolio value due to changes in the shape of the benchmark yield curve rather than its level. Effective duration nevertheless remains the standard measure, because parallel shifts explain more than 75% of the variation in bond portfolio returns.

Key rate duration

Key rate duration is a more precise method of quantifying bond price sensitivity to interest rates and is superior to effective duration for measuring the impact of nonparallel yield curve shifts. It is the sensitivity of the value of a security or bond portfolio to changes in a single par rate, holding all other par rates constant, so it isolates price sensitivity to a change in the yield at a particular maturity only. Numerically it is the approximate percentage change in the value of a bond portfolio in response to a 100 basis point change in the corresponding key par rate, all other par rates held constant. Every security or portfolio has a set of key rate durations, one for each key rate, and their sum is the effective duration of the portfolio — a portfolio with D1 = 0.7, D5 = 3.5 and D25 = 9.5 has an effective duration of 13.7. The model is ΔP/P ≈ −D1Δr1 − D5Δr5 − D25Δr25.

Level, steepness and curvature

An alternative to decomposing yield curve risk into sensitivity to changes at various maturities is to decompose it into sensitivity to three categories of yield curve movement. Level (ΔxL) is a parallel increase or decrease of interest rates. Steepness (ΔxS) is long-term interest rates increasing while short-term rates decrease. Curvature (ΔxC) — increasing curvature — means short- and long-term interest rates increase while intermediate rates do not change. It has been found that all yield curve movements can be described using a combination of one or more of these movements. The portfolio model is ΔP/P ≈ −DLΔxL − DSΔxS − DCΔxC, where DL, DS and DC are the portfolio’s sensitivities to changes in the curve’s level, steepness and curvature.

Maturity structure of yield curve volatilities

Interest rate volatility is a key concern for bond managers because it drives price volatility in a fixed income portfolio, and it becomes particularly important when securities have embedded options, which are especially sensitive to volatility. The term structure of interest rate volatility is the graph of yield volatility versus maturity, with volatility at time t for a security of maturity T denoted σ(t,T). Short-term interest rates are generally more volatile than long-term rates. Volatility at the long-maturity end is thought to be associated with uncertainty regarding the real economy and inflation, while volatility at the short-maturity end reflects risks regarding monetary policy.

Yield curve shifts and investor actions

Shifts are classified on two axes, direction and shape, producing four categories: bullish steepening, bullish flattening, bearish steepening and bearish flattening, where bullish denotes falling rates. In expectation of a rise in rates investors will lower the duration of their bond portfolios, and in expectation of a fall they will extend it. Expectations of a steepening of the yield curve may lead investors to go long short-term bonds and short longer-term bonds. Such trades may be designed to be duration-neutral, so that a change in the level of interest rates does not affect the value of the portfolio and the position is a pure bet on curve shape. Investors with long-only mandates instead rotate between a bullet portfolio, concentrated in a single maturity, and a barbell portfolio, holding short and long maturities. An investor expecting a bullish flattening may rotate out of a bullet portfolio and into a barbell.