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# Matrix pricing

When bonds are not yet issued or where they are not actively traded, their yield and price are determined using comparable bonds, bonds whose time to maturity, coupon rate, and credit quality are similar, through a process called matrix pricing.

## Steps in matrix pricing

The following steps outline the process used in matrix pricing:

STEP 1: Identify comparable bonds having the same credit quality and calculate their yield to maturity.

STEP 2: Find out average yield for all bonds of the same maturity.

STEP 3: Use linear interpolation to work out estimated yield for the bond being valued.

$Estimated\ Yield=Y_S+\frac{T_i-T_S}{T_L-T_S}\times(Y_L-Y_S)$

Where YS is the yield on the shorter maturity bond, YL is the yield on longer maturity bond, Ti is the maturity of the bond being valued, TL is the maturity of the longer maturity bond and TS is the maturity of shorter maturity bonds.

STEP 4: Price the non-traded bond using the estimated yield identified above.

## Example

If we need to work out the yield to maturity of a 3-year bond paying 4% annual coupon and we have a 2-year bond with a yield of 2.5% and 4-year bond with a yield of 3.5%, the yield of a 3-year bond can be interpolated as follows:

$Estimated\ Yield=2.5\%+\frac{3-2}{4-2}\times(3.5\%-2.5\%)=3\%$

The bond price can be worked out using the bond pricing formula.

Matrix pricing is also used to find out the appropriate spread for a bond over the benchmark rate, which is typically the yield on government bonds.

For example, assume an issuer wants to float a 5-year bond and it currently has a 4-year bond with a required yield of 3% but the benchmark bonds are available only 3- and 5-year maturities having a yield of 2% and 2.4% respectively. We can use matrix pricing to find the estimated benchmark yield for the 4-year maturity would be 2.2% (average of 2% and 2.4%), work out the required spread (80 bps i.e. excess of 3% over the benchmark yield of 2.2%) and then use the term structure of yield spread to see how the 5-year spread would differ from 4-year spread and price its 5-year bond accordingly.