Questions:
Imagine two bonds. One is a 3.50% annual coupon bond which is currently trading at 102.55 in the market with a maturity of three years.  The other bond is a 7.25% annual coupon bond which is currently trading at 122.4 and has a maturity of 5 years. What is the YTM of these bonds?

Our investment banking firm is in the process of underwriting a corporate bond which will be issued for four years at an annual coupon of 4.5%. We are trying to price the bond according to the market. Since this is a new issue, there is no straightforward way to price this bond. 

However, those two bounds are very similar and were underwritten by our investment bank earlier. They are actively traded on the market and have similar credit risk and features as the one being underwritten.

I want you to use these two bonds to provide a good estimate of the price at which we should underwrite our client's bond.

Reference answers:
YTM of a bond can be calculated using the following formula

P = C/(1+r)^-1 + C/(1+r)^-2 + ... + (C+F)/(1+r)^-n
where p is the price at which the bond is currently trading, C is the coupon payment on the bond, F is the future value at which the bond will be redeemed at maturity, n is the number of periods from issue of the bond to its redemption. r is the YTM which we will find out.

Bond 1
102.55 = 3.5/(1+r)^-1 + 3.5/(1+r)^-2 + 103.5/(1+r)^-3
r = 2.605%

Bond 2
-122.4 = 7.25(1+r)^-1 +7.25(1+r)^-2 +7.25(1+r)^-3 +7.25(1+r)^-4 +107.25(1+r)^-5
r = 2.437%

The first bond has a yield to maturity of 2.605% and the second bond has a yield to maturity of 2.437%.

This can be achieved using Matrix pricing as we have access to the two comparable bonds. Matrix pricing involves using prices of comparable, actively traded bonds with similar structure, coupon rates, credit rating, and similar maturities to estimate the price of a new bond with no market (CFI, 2024).

The problem is that we do not have exactly similar bond but we do have comparable bonds with maturities of 3 and 5 years so we can value the new bond (maturity of 4 years) by taking the average of the YTMs of these two bonds. 

Estimating the YTM using the two bonds:

r = (2.605% + 2.437%) / 2 = 2.521%

Estimating the Bond Price at which the bond should be underwritten

Value of bond = 4.5(1+0.02521)^-1 + 4.5(1+0.02521)^-2 + 4.5(1+0.02521)^-3 + 104.5(1+0.02521)^

Value of bond = $107.44

Based on the comparable bonds in the market, the matrix pricing suggests that the new bond should be underwritten at $107.44.