Question

The face value of a bond is $1,000. The bonds have a 4.25% coupon rate paid semi-annually and mature in six years. What is the yield to maturity (express at an annual rate) for the bonds if an investor buys them at the $875 market price?

Answer 1

Semi - annual interest payment = $1000 x 4.25% x 6 / 12 = $21.25

Total semi - annual periods till maturity = 6 x 2 = 12

First, compute approximate YTM using the following formula -

where, I = interest payment, RV = redeemable value, MV = market value, n = no. of time periods

Approx YTM = [ $21.25 + { ($1000 - $875) / 12} ] / [ (1000+875) / 2 ] = 0.033777777

or, Approx YTM = 3.38%

Now, YTM is close to this rate. We need to choose two rates close to approximate YTM and compute the market value of the bond at those rates. The rate at which market value is equal to the offered value will be the YTM. Remember, the closer the rates to YTM, the closer will be your answer.

Lets take 3.35% and 3.45%

At 3.35%, Bond price = $21.25 x PVIFA (3.35%, 12) + $1000 x PVIF (3.35%, 12) = $21.25 x 9.74920205832 + $1000 x 0.67340173096 = $880.572274699

At 3.45%, Bond price = $21.25 x PVIFA (3.45%, 12) + $1000 x PVIF (3.45%, 12) = $21.25 x 9.69183196617 + $1000 x 0.66563179712 = $871.583226401

Now, we need to interpolate -

Difference required = $880.572274699 - $875 = $5.572274699

Total difference = $880.572274699 - $871.583226401 = $8.989048298

YTM = Lower rate + Difference in rates x (Difference required / Total difference)

or, YTM = 3.35% + 0.10% x ($5.572274699 / $8.989048298) = 3.411989859571% or 3.41%

Now, this is the semi - annual rate, therefore, annual YTM = 3.41% x 2 = 6.82%

Note :

PVIF = 1 / (1 + r)ⁿ