When fixed-rate bonds are not actively traded or there is no market price to calculate the rate of return required by investors, it is common to estimate the market discount rate and price based on quoted or flat prices of more frequently traded comparable bonds with similar credit quality, maturity, and coupon rates. This estimation process is called matrix pricing.

## Illustrative Example

Assume that an analyst needs to value a 4-year, 5% annual coupon payment bond, Bond-K, which is not actively traded. The analyst could find some other bonds, however, with a comparable credit quality:

- Bond-L: 3-year, 5.5% annual coupon payment bond at a price of 108; and
- Bond-M: 5-year, 4.5% annual coupon payment bond at a price of 105.

Therefore, we start off with the following matrix:

$$

\begin{array}{l|c|c|c}

\text{} & \begin{array}{c} \text{4.5%-coupon} \\ \text{Bond} \end{array} & \begin{array}{c} \text{5%-coupon} \\ \text{Bond} \end{array} & \text{5.5%-coupon Bond} \\

\hline

\text{3-year Bond} & \text{} & \text{} & \begin{array}{c} \text{Price = 108} \\ \text{YTM = ?} \end{array} \\

\hline

\text{4-year Bond} & \text{} & \begin{array}{c} \text{Price = ?} \\ \text{YTM = ?} \end{array} & \text{} \\

\hline

\text{5-year Bond} & \begin{array}{c} \text{Price = 105} \\ \text{YTM = ?} \end{array} & \text{} & \text{} \\

\end{array}

$$

### Step 1: Find the Yield-to-maturity of Observed Bonds

Using the financial calculator for Bond-L: N=3; PV=-108; PMT=5.5; FV=100; CPT => I/Y = 2.6887. The required yield on Bond-L is 2.69%.

For Bond-M: N=5; PV=-105; PMT=4.5; FV=100; CPT => I/Y = 3.3959. The required yield on Bond-M is 3.40%:

We now have two more data points in our matrix:

$$

\begin{array}{l|c|c|c}

\text{} & \begin{array}{c} \text{4.5%-coupon} \\ \text{Bond} \end{array} & \begin{array}{c} \text{5%-coupon} \\ \text{Bond} \end{array} & \text{5.5%-coupon Bond} \\

\hline

\text{3-year Bond} & \text{} & \text{} & \begin{array}{c} \text{Price = 108} \\ \text{YTM = 2.69%} \end{array} \\

\hline

\text{4-year Bond} & \text{} & \begin{array}{c} \text{Price = ?} \\ \text{YTM = ?} \end{array} & \text{} \\

\hline

\text{5-year Bond} & \begin{array}{c} \text{Price = 105} \\ \text{YTM = 3.4%} \end{array} & \text{} & \text{} \\

\end{array}

$$

### Step 2: Find the Yield-to-maturity of the Non-actively Traded Bond

The estimated market discount rate for a 4-year bond having the same credit quality is the average of two required yields:

$$ YTM_{Bond-K} = \frac{2.69\% + 3.40\%}{2} = 3.045\% $$

We now have the following matrix:

$$

\begin{array}{l|c|c|c}

\text{} & \begin{array}{c} \text{4.5%-coupon} \\ \text{Bond} \end{array} & \begin{array}{c} \text{5%-coupon} \\ \text{Bond} \end{array} & \text{5.5%-coupon Bond} \\

\hline

\text{3-year Bond} & \text{} & \text{} & \begin{array}{c} \text{Price = 108} \\ \text{YTM = 2.69%} \end{array} \\

\hline

\text{4-year Bond} & \text{} & \begin{array}{c} \text{Price = ?} \\ \text{YTM = 3.045%} \end{array} & \text{} \\

\hline

\text{5-year Bond} & \begin{array}{c} \text{Price = 105} \\ \text{YTM = 3.4%} \end{array} & \text{} & \text{} \\

\end{array}

$$

### Step 3: Find the Price of the Non-actively Traded Bond

Given an estimated yield-to-maturity of 3.045%, the estimated price of the 4-year 5% illiquid bond is 107.26 per 100 of each par value. To find this value, we need to plug in the following variables into the financial calculator: N=4; I/Y=3.045; PMT=5; FV=100; CPT => PV = -107.26

Alternatively, we could use a timeline to find the same value:

$$

\begin{array}{l|ccccc}

\text{Time Period} & 1 & 2 & 3 & 4\\

\hline

\text{Calculation} & \frac {\$5}{{\left(1+3.045\%\right)}^{ 1 } } & \frac { \$5 }{ { \left( 1+3.045\% \right) }^{ 2 } } & \frac { \$5 }{ { \left( 1+3.045\% \right) }^{ 3 } } & \frac { \$105 }{ { \left( 1+3.045\% \right) }^{ 4 } } \\

\hline

\text{Cash Flow} & \$4.85 & +\$4.70 & +\$4.56 & +\$92.92 & =\$107.03 \\

\end{array}

$$

### Benefits of Matrix Pricing

Matrix pricing is also used during the underwriting process to get an estimate of the required yield-spread over the benchmark rate, where the benchmark rate is generally the yield-to-maturity of a similar government bond. Yield spreads are often stated in terms of basis points (bps). For example, if we have a yield-to-maturity of 3.75% and the comparable government bond yield (or benchmark rate) is 2.5%, the yield-spread will be 1.25% or 125 bps.

## Question

Matrix pricing allows investors to estimate:

A. The required yield spread as well as prices for bonds that become substantially risky after bond’s initial issuance

B. The bond coupon rates for bonds that are comparable to government bonds

C. The market discount rates as well as the prices for bonds that are not actively traded

SolutionThe correct answer is C.

Matrix pricing is a price estimation process that uses market discount rates based on the quoted prices of similar bonds (similar maturity, coupon rates, and credit quality) when a fixed-rate bond is not actively traded or there is no market price.

*Reading 44 LOS 44e: *

*Describe matrix pricing*