# Yield Spread

Yield spread is the difference between the yield to maturity on different debt instruments. Common examples of yield spreads are g-spread, i-spread, zero-volatility spread and option-adjusted spread.

Bond yield is the internal rate of return of the bond cash flows. It is the rate of return that a bondholder earns if he holds the bond till maturity and receive all the cash flows at the promised dates. In an efficient market, bond yield is a barometer that can be used to asset the risk inherent in the bond. In general, higher yields means the bond has higher risk and hence lower price.

When yield on one bond or class of bonds is compared with another, the difference is called the yield spread. It helps us dissect the different risks related to bonds such as credit risk, interest rate risk, inflation risk, etc. For example, if we compare the yield on regular US Treasury bonds with yield on Treasury Inflation Protected Securities (TIPS) of the same maturity, the yield spread represent the inflation risk. If we match maturities, regular Treasury Bonds and TIPS are identical in all aspects except the inflation protection. TIPS have inflation protection, but Regular Treasury Bonds do not have, hence, the difference in yield i.e. the yield spread represent the inflation risk. In the same way, we can use yield spread to study the factors that drive bond prices.

## Types

### G-spread

G-spread (also called nominal spread) is the difference between yield on Treasury Bonds and yield on corporate bonds of same maturity. Because Treasury Bonds can be assumed to have zero default risk, the difference between yield on corporate bonds and Treasury bonds represent the default risk.

$$ G\ Spread=Y_c-Y_g $$

Where Yc is the yield on non-treasury bond and Yg is the yield on government bond of the same maturity.

### I-spread

I-spread stands for interpolated spread. It is the difference between yield on a bond and the swap rate, i.e. the interest rate applicable to the fixed leg in the floating-for-fixed interest rate swap. The difference between yield on a bond and a benchmark curve such as LIBOR is useful in assessing credit risk of different bonds. Higher i-spread means higher credit risk. I-spread is typically lower than the G-spread.

### Z-spread

Z-spread stands for zero-volatility spread. It is the spread that must be added to each spot interest rate to cause the present value of the bond cash flows to equal the bond’s price.

While G-spread and I-spread just measure the difference between the static yield to maturity of the bond and the Treasury yields or benchmark rate, Z-spread determines the difference in yields with reference to whole term structure of interest rates.

Z-spread can be calculated by solving the following equation for Z:

$$ P=\frac{{\rm CF}_1}{{({1+S}_1+Z)}^1}+\frac{{\rm CF}_2}{{({1+S}_2+Z)}^2}+...+\frac{{\rm CF}_n}{{(1+S_n+Z)}^n} $$

Where P is the price of the bond, CF_{1}, CF_{2} and CF_{n} are the first, second and nth cash flows, S_{1}, S_{2} and S_{n} are the first, second and nth spot interest rate and Z is the zero-volatility spread.

### Option-Adjusted Spread (OAS)

Option-adjusted spread equals zero-volatility spread minus the value of call option as stated in basis points. It is the appropriate yield measure for a callable bond:

$$ OAS=Z\ Spread\ -\ Option\ Value $$

## Example

If the 2-year Treasury bond yield is 2.25% and 2-year LIBOR swap rate is 2.69%, determine the G-spread and I-spread on a bond with 2 years to maturity yielding 3.5%.

The bond has a par value of $1,000, trades at 99% of its face value and pays annual coupon payments based a 3.4% coupon rate. Calculate the zero-volatility spread if the 1-year and 2-year treasury yield is 2.14% and 2.42%.

G-spread just equals the difference between the bond yield and the Treasury yield. In this case it equals 1.25% (=3.5% - 2.25%).

I-spread equals the difference between bond yield and swap rate. It equals 0.81% (=3.5% - 2.69%).

Given the bond price of $990, annual coupon payments of $34 (=$1,000 × 3.4%) and Treasury spot rates for 1 and 2 years of 2.14% and 2.42%, we can work out z-spread by solving the following equation for Z:

$$ $990=\frac{$34}{{(1+2.14\%+Z)}^1}+\frac{$34\ +$1,000}{{(1+2.42\%+Z)}^2} $$

The above equation can be solved using hit-and-trial method or using Excel Goal Seek. Z equals 1.51%.