Question

Consider a​ five-year, default-free bond with annual coupons of 6% and a face value of $1,000 and assume​ zero-coupon yields on​ default-free securities are as summarized in the following​ table:

Maturity	1 year	2 years	3 years	4 years	5 years
​Zero-Coupon Yields	5.00​%	5.30​%	5.50​%	5.70​%	5.80​%

a. What is the yield to maturity on this​ bond? The yield to maturity on this bond is ............% ​(Round to three decimal​ places.)

b. If the yield to maturity on this bond increased to 6.20 %​, what would the new price​ be?

The new price would be ​$ ............. ​ (Round to the nearest​ cent.)

Answer 1

Given:
Coupon rate = 6% PA
Face Value = $1000
Interest = $1000*6% = $60

a) Calculation of YTM of the Bond
Step 1: Calculation of Current Price of Bond

In order to calculate the YTM of the bond, we first need to calculate the Current Price of the Bond.
As per the given zero-coupon yields on​ default-free securities, the Current Price of this bond will be calculated as follows:

Where CFt = cash flow at period t
rt = zero-coupon yield on default-free securities at year t

Bo = 60 / (1+0.05)^1 + 60 / (1+0.053)^2 +60 / (1+0.055)^3 + 60 / (1+0.057)^4 + (60+1,000) / (1+0.058)^5
Bo = 60 * 0.95238095 + 60 * 0.90186858 +60 * 0.85161366 + 60 * 0.8011246 + 1,060 * 0.75434785
Bo = $1,010.03

Step 2 : Calculation of YTM of Bond

Using the Above Equation.

1,010.03 = 60 / (1+ r)^1 + 60 / (1+r)^2 +60 / (1+ r)^3 + 60 / (1+ r)^4 + 1,060 / (1+r)^5
where r = Yield to Maturity

Using the trial and error method.
(i) Let YTM = 5%.

Bo = 60 / (1+0.05)^1 + 60 / (1+0.05)^2 +60 / 1+0.05)^3 + 60 / (1+0.05)^4 + (60+1,000) / (1+0.05)^5
Bo = $ 1,043.29
Since, at YTM = 5%, Bond Value is greater than the Current Bond Price we need to use a higher YTM rate

(ii) Let YTM = 6%

Bo = 60 / (1+0.06)^1 + 60 / (1+0.06)^2 +60 / 1+0.06)^3 + 60 / (1+0.06)^4 + (60+1,000) / (1+0.06)^5
Bo = $ 1,000

Since, at YTM = 6%. The value of Bond is less than the Bond Current Price. Therefore, it means YTM lies between 5% to 6%

Using Interpolation

YTM = 5% + (6% -5%) * ( 1,043.29 - 1,010.03) / (1,043.29 - 1000)
YTM = 5% + (1%) * 33.26 / 43.29
YTM = 5% + 0.768378
YTM = 5.768378 % or 5.77% (Approx)

b) Calculation of Bond Price at YTM = 6.20%

Bo = 60 / (1+0.062)^1 + 60 / (1+0.062)^2 +60 / (1+0.062)^3 + 60 / (1+0.062)^4 + (60+1,000) / (1+0.062)^5
Bo = 60 * 0.94161959 + 60 * 0.88664744 +60 * 0.8348846 + 60 * 0.78614369 + 1,060 * 0.7402483
Bo = $ 991.62

Bond Price at YTM = 6.20% is $ 991.62