G-spread, I-spread, Z-spread and option-adjusted spread

What is a benchmark, and what is a spread?

A bond yield has two parts. The benchmark is a reference rate close to risk-free: the expected real rate plus expected inflation. It captures the top-down, macroeconomic forces that move every bond. The yield spread is what is left: yield to maturity minus the benchmark. It captures the bottom-up forces specific to the issuer and the bond.

Yield to maturitybenchmark plus spread

• Benchmark rate Memorize top-down, macroeconomic; moves every bond in the market
  • Expected real rate the reward for waiting
  • Expected inflation compensation for lost purchasing power
• Spread Memorize bottom-up, specific to the issuer and the bond
  • Credit risk chance and cost of default
  • Liquidity cost of trading it
  • Taxation tax treatment of the interest

$$\text{yield spread} = YTM - \text{benchmark yield}$$

Splitting a yield this way shows why a price moved. If yields rose across the market, the benchmark moved. If only this issuer’s bond cheapened, its spread widened.

The usual benchmark is a government bond. The on-the-run bond is the most recently issued for its maturity. It is the most actively traded, and its coupon sits close to the market rate, so it prices near par. Seasoned bonds are off-the-run . On-the-run bonds typically yield a little less than off-the-run bonds of similar maturity, because demand is greater and they can cost less to finance in the repo market.

The spread over one particular benchmark bond is a benchmark spread . It pays for credit and liquidity risk, and possibly the tax treatment of the bond.

How do you calculate the G-spread and I-spread?

The G-spread is the bond’s yield minus the government yield at the same maturity. Governments rarely have a bond maturing on exactly the right day, so you interpolate between the two nearest.

Yield spread measuresall are yield minus a benchmark; they differ in the benchmark

• G-spread Memorize over a government yield, interpolated to the bond’s maturity
• I-spread Memorize over the swap rate of the same currency and tenor
• Z-spread Memorize constant spread over every point of the spot curve
• Option-adjusted spread Memorize Z-spread less the value of the embedded option
  • Callable bond OAS below the Z-spread

$$y_{gov} = y_1 + \frac{T - T_1}{T_2 - T_1}\,(y_2 - y_1), \qquad G = YTM - y_{gov}$$

The weights run toward the nearer maturity: a bond closer to $T_1$ takes most of its government yield from $y_1$.

The I-spread uses a swap rate in place of a government yield: the standard swap rate in the same currency and with the same tenor. It is the usual convention for euro-denominated corporate bonds, which are priced as “mid-swaps plus x bps”, where mid-swaps is the average of the bid and offered swap rates. Issuers use the I-spread to compare a fixed-rate bond with floating-rate funding, such as a bank loan, and investors read it as a measure of credit risk.

$$I = YTM - \text{swap rate (same currency, same tenor)}$$

An asset swap converts the fixed coupon of a particular bond into a floating market reference rate plus or minus a spread. For a bond priced near par, that spread approximates the price of the bond’s credit risk over the reference rate.

Go deeper Why an I-spread can be bigger than a G-spread

Swap rates and government yields are different curves. When government yields sit below swap rates, as in the example, a bond has a smaller spread over governments than over swaps. When they sit above, the G-spread is the larger one. Neither is more correct. They answer different questions, so you track each against its own history.

What is the Z-spread?

The G-spread and I-spread compare two single numbers, so they discount every cash flow at one yield. A curve is not flat, so that is crude. The Z-spread uses the whole spot curve. It is the one constant spread $Z$ that, added to every benchmark spot rate, makes the present value of the bond’s cash flows equal its price.

$$PV = \frac{PMT}{(1 + z_1 + Z)^{1}} + \frac{PMT}{(1 + z_2 + Z)^{2}} + \cdots + \frac{PMT + FV}{(1 + z_N + Z)^{N}}$$

Here $z_t$ are the benchmark spot rates, derived from the government (or swap) yield curve, which the term structure module covers in full. Read that first if spot rates are new to you. The Z-spread is also called the static spread, because it is constant, and the zero-volatility spread. It is usually found with a solver or by trial and error.

Exam tip The exam usually gives you the Z-spread and asks for the price, which is one pass through the formula. Finding Z from a price is a trial-and-error job, so expect to be given the answer options.

What is the option-adjusted spread?

A callable bond pays the investor extra for a risk: the issuer can call it when rates fall. The Z-spread therefore includes compensation for the option the investor has sold. The option-adjusted spread takes it out. An option model and an interest rate volatility assumption give the value of the call in basis points a year.

$$OAS = Z\text{-spread} - \text{option value (bps a year)}$$

For a callable bond the option value is positive, so the OAS is lower than the Z-spread. What remains is the pay for credit and liquidity risk.

Go deeper Putable bonds and the sign of the option

For a putable bond the option favors the investor, so the adjustment runs the other way and the OAS is above the Z-spread. The idea is the same: OAS removes the option’s effect from the Z-spread. The Level I exam concentrates on callable bonds, where OAS is lower. Option pricing models and volatility assumptions come at Level II.